Geometric symbol calculusfor pseudodifferential operators

Vladimir Sharafutdinov ∗

April 2001 — August 2002; Seattle — Novosibirsk

Abstract

A connection on a manifold allows us to define the full symbol of a pseudod-ifferential operator in an invariant way. The latter is called the geometric symbolto distinguish it from the coordinate-wise symbol. The traditional calculus is de-veloped for geometric symbols: an expression of the geometric symbol through thecoordinate-wise one, formulas for the geometric symbol of the product of two oper-ators, and of the dual operator.

Key words: Pseudodifferential operators, Connections on manifolds, Covariant deriva-tive.AMS Subject Classification: 35S05, 53C05.

1 Introduction

Some rather elementary part can be distinguished in the theory of pseudodifferentialoperators which deals with the representation of symbols of operators by asymptotic series.We call this part the symbol calculus. It includes, first of all, the theorem of transformingthe symbol of an operator under a coordinate change, formulas for the symbols of thedual operator and of the product of two operators. This part is very important becausethe symbol calculus underlies in many other results of the theory.

While dealing with pseudodifferential operators on domains of the Euclidean space,there is a one-to-one correspondence between operators (modulo smoothing operators)and their full symbols (modulo rapid decay symbols). The situation is quite differentwhen we consider pseudodifferential operators on manifolds. Only the principal symbolof an operator can be invariantly defined in the latter case. As has been first noticedby H. Widom, the situation can be improved by introducing a connection on a manifold.The full symbol of a pseudodifferential operator can be defined in an invariant way on amanifold endowed with a connection. The corresponding symbol calculus was developedby H. Widom [7, 8] and later by Yu. Safarov [?] and M. Pflaum [5]. Unfortunately,the main formulas of the calculus are presented in a rather cumbersome form in these

∗Supported by CRDF, Grant RM2–2242; and by NSF, Grant DMS–9765792.

1

papers which is not comfortable for usage. These formulas are not of completely invariantcharacter as using special coordinates, special representation of phase functions and soon.

Let us briefly discuss the question: what is an invariant formula? The modern math-ematical style presumes that invariant (independent of the choice of coordinates) notionsare introduced by invariant definitions and invariant relations should be written in acoordinate free form. It is very difficult, if possible, to take this view while treatinghigher-order (pseudo)differential operators. Therefore we adhere to the following view-point in the present paper which is still popular in physics: a formula is an invariant oneif it keeps its form under a coordinate change.

The present paper is formally independent of the above-mentioned articles. Startingwith a few definitions, we develop the geometric version of the theory of pseudodifferentialoperators on manifolds endowed with a connection. The main results are obtained in aninvariant form in the above sense. This is achieved, first of all, by using the adequatetensor analysis machinery.

Expedience of introducing a connection while treating pseudodifferential operators canbe demonstrated by the following challenging example. One of the first and most impor-tant ingredients of the theory of pseudodifferential operators is the theorem of transform-ing the symbol of an operator under a coordinate change. We recall the theorem herefollowing [4] but with slightly different notations.

Let κ : X → X ′ be a diffeomorphism between two open sets of Rn. For (x, ξ) ∈ X×Rn,denote the corresponding point of X ′×Rn by (x′, ξ′), i.e., x′ = κ(x), ξ′ = (tκ′(x))−1ξ. LetA ∈ Ψm

cl (X) be a classical pseudodifferential operator, and A′ = (κ∗)−1 Aκ∗ ∈ Ψmcl (X

′).Denote the symbols of A and A′ by a(x, ξ) and a′(x, ξ) respectively. The asymptoticexpansions

a(x, ξ) ∼ a0(x, ξ) + a1(x, ξ) + a2(x, ξ) + . . . ,

a′(x, ξ) ∼ a′0(x, ξ) + a′1(x, ξ) + a′2(x, ξ) + . . .

of the symbols are related by the following transformation formulas:

a′0(x′, ξ′) = a0(x, ξ), (1.1)

a′1(x′, ξ′) = a1(x, ξ)− i

2

∂2x′p

∂xj∂xkξ′p

v

∇jv

∇ka0(x, ξ), (1.2)

a′2(x′, ξ′) = a2(x, ξ)− i

2

∂2x′p

∂xj∂xkξ′p

v

∇jv

∇ka1(x, ξ)− 1

6

∂3x′p

∂xj∂xk∂xlξ′p

v

∇jv

∇kv

∇la0(x, ξ)

− 1

8

∂2x′p

∂xj∂xk

∂2x′q

∂xl∂xmξ′pξ

′q

v

∇jv

∇kv

∇lv

∇ma0(x, ξ), (1.3)

and so on. Here the notationv

∇j = ∂/∂ξj is used. The operatorv

∇ is called the verticalderivative because it is the gradient in the fibers of the cotangent bundle X ×Rn → X.We use the rule: the summation from 1 to n is assumed whenever upper and low indicesare repeated in a monomial. The length of the corresponding formula for a′j grows veryfast with j.

2

Formula (1.1) means that the principal symbol

a0(x, ξ) = a0(x, ξ) (1.4)

is invariant under a coordinate change. It is just this formula that allows us to define theprinciple symbol as a function on the cotangent bundle T ∗X in the case of a pseudodif-ferential operator on an abstract manifold X.

Now, look at formula (1.2). The function a1(x, ξ) is not invariant in view of thepresence of the second order derivatives ∂2x′p/∂xj∂xk in the formula. There is a wellknown way for treating such noninvariance in geometry. To this end let us introducea symmetric connection ∇ on X. In local coordinates, a connection is given by theChristoffel symbols Γl

jk = Γlkj which are transformed under a coordinate change according

to the formula

Γljk =

∂x′p

∂xj

∂x′q

∂xk

∂xl

∂x′rΓ′rpq +

∂xl

∂x′p∂2x′p

∂xj∂xk. (1.5)

Expressing the second order derivative ∂2x′p/∂xj∂xk from the latter equality and substi-tuting the expression into (1.2), we obtain after simple transformations

a′1(x′, ξ′)− i

2Γ′pjk(x

′)ξ′p∂2a′0(x′, ξ′)

∂ξ′j∂ξ′k= a1(x, ξ)− i

2Γp

jk(x)ξp∂2a0(x, ξ)

∂ξj∂ξk

.

This means that the function

a1(x, ξ) = a1(x, ξ)− i

2Γp

jk(x)ξp

v

∇jv

∇ka0(x, ξ) (1.6)

is invariant, i.e., it is transformed under a coordinate change according to the same rule(1.1) as the principal symbol a0(x, ξ) = a0(x, ξ). Therefore we can consider a1(x, ξ) asa well defined function on the cotangent bundle T ∗X in the case of a pseudodifferentialoperator on a manifold X endowed with a connection.

It turns out this procedure can be extended to higher order symbols aj(x, ξ) for allj. We can thus define invariant functions aj(x, ξ) on T ∗X as well as the full symbola ∼ a0 + a1 + a2 + . . . . For instance the formula for a2 is as follows:

a2 = a2− i

2Γp

jkξp

v

∇jv

∇ka1− 1

6

(∂Γp

jk

∂xl+ Γp

jqΓqkl

)ξp

v

∇jv

∇kv

∇la0− 1

8Γp

jkΓqlmξpξq

v

∇jv

∇kv

∇lv

∇ma0.

(1.7)To distinguish between a(x, ξ) ∼ a0 + a1 + . . . and a(x, ξ) ∼ a0 + a1 + . . . , we call a(x, ξ)the coordinate-wise symbol of the operator A (with respect to a given coordinate system),and call a(x, ξ) the geometric symbol of A (with respect to a given connection).

The above approach seems to be hardly applicable for finding the geometric symbolsaj for j > 2 because the corresponding formulas are very cumbersome. Therefore we useanother approach in the main part of the article. Starting with a function a(x, ξ) on thecotangent bundle T ∗X of a manifold X endowed with a connection ∇, we will give aninvariant definition of the pseudodifferential operator A = a(x,−i∇) whose full geometricsymbol is the given function. After this definition, formulas like (1.4) and (1.6)–(1.7) canbe considered as the expression for the geometric symbol in local coordinates.

3

A remark on the style of the present article is in order. Main ideas are first discussedfor differential operators, then for scalar pseudodifferential operators, and finally the mostgeneral case of pseudodifferential operators is considered in the last section. This stylemakes the the presentation rather long but, as the author hopes, it makes reading easier.

The theory of pseudodifferential operators must include differential operators a(x,−i∇)as a special case of symbols a(x, ξ) depending polynomially on ξ. The theory of such differ-ential operators has some specifics because the covariant derivatives −i∇j do not commuteunlike the case of partial derivatives Dj = −i∂/∂xj. Therefore we start with discussingdifferential operators in Section 2 that contains also the presentation of some notionsof tensor analysis. Probably, the horizontal covariant derivative is the most importantand less known tensor machinery which we need. We present only the definition of thehorizontal derivative referring the reader to Chapter 3 of [6] for details.

In Section 3, we define pseudodifferential operators on a manifold endowed with aconnection. This definition coincides with the classical one in the case of a flat connection.We demonstrate that the symbol can be recovered from the operator up to a smoothingterm.

Section 4 is devoted to representing a pseudodifferential operator in local coordinates.We show that formulas like (1.4) and (1.6)–(1.7) can be, in principle, written down foraj(x, ξ) with any j.

The formula for the symbol of the product of two operators is proved in Section 5,and the dual operator is considered in Section 6.

Sections 7 and 8 consider operators on vector bundles. The principal thing here isthat, to introduce a geometric symbol, we need not only a connection on the manifoldbut also a connection on the bundle. Most proofs here follow the same line as in thescalar case. Therefore we do not present any proof in detail but only discuss necessarymodifications.

The central result of the paper is the formula for the symbol of the product of twooperators. Theorems 2.4, 5.1, 7.6, and 8.3 are devoted to the formula. Theorem 2.4 statesthe formula for scalar differential operators, and Theorem 5.1 generalizes the formula toscalar pseudodifferential operators. Theorems 7.6 and 8.3 state the formula for differentialand pseudodifferential operators on vector bundles. It is remarkable that we have in factthe same formula in each of these cases. The only difference between the scalar and vectorbundle cases is in the definition of the coefficients of the formula. The coefficients aredetermined by the curvature tensor of the connection on the manifold in the scalar case,while the coefficients depend also on the curvature tensor of the bundle connection in thecase of vector bundles. In both the cases our definition gives an algorithm for computingthe coefficients, but the volume of calculations grows very fast with the coefficient number.The problem of creating a software for computing the coefficients seems to be of greatimportance, but it is still open. Our proofs of these four theorems do not duplicate eachother. There is the following interesting interaction of the proofs. On one hand, whileproving the formula for differential operators, we obtain an explicit expression for thecoefficients of the formula. At the same time we cannot obtain an estimate on degreesof coefficients which guarantees the convergence of the asymptotic series. On the otherhand, while proving the formula for pseudodifferential operators, we obtain the estimatebut cannot get an explicit formula for coefficients. Combining these two cases we get

4

both, the estimate and the explicit formula.Comparing with the above-mentioned predecessors, the present paper is more close

to the article [?] by Yu. Safarov. At the same time the following principal distinctionbetween our approach and Safarov’s one should be mentioned. Safarov gives an invariantdefinition of the Schwartz kernel of a pseudodifferential operator. To this end he has toconsider operators on densities of different orders. In our approach, the Schwartz kerneldoes not participate explicitly in the definition of the symbol, but the exponential mapand the parallel transport along geodesics are the main ingredients of the definition. Thisdistinction can be demonstrated by the example of the Laplace — Beltrami operator ∆ ona Riemannian manifold. According to our definition, the full geometric symbol a(x, ξ) of∆ coincides with the principal symbol and is a(x, ξ) = −‖ξ‖2 = −gij(x)ξiξj. By Safarov,the full geometric symbol of ∆ is −‖ξ‖2 − S(x)/3, where S is the scalar curvature. Suchdistinctions can be essential in applications.

2 Differential operators

Given a manifold X, we denote by τX = (TX, π,X) (by τ ∗X = (T ∗X, π, X)) the tangent(cotangent) bundle of X. The points of TX (of T ∗X) will be denoted by the pairs (x, v)(the pairs (x, ξ)) where x ∈ X and v ∈ TxX (ξ ∈ T ∗

xX). By C∞(τ rs X) we denote the space

of smooth complex-valued tensor fields on X which are r times contravariant and s timescovariant. Such a tensor field u ∈ C∞(τ r

s X) is determined in the domain U of a localcoordinate system (x1, . . . , xn) by its coordinates uj1...jr

k1...ks∈ C∞(U) which are transformed

by the formula

u′j1...jr

k1...ks=

∂x′j1

∂xl1. . .

∂x′jr

∂xlr

∂xm1

∂x′k1. . .

∂xms

∂x′ksul1...lr

m1...ms(2.1)

under a coordinate change. In particular, C∞(τ 00 X) = C∞(X) is the algebra of smooth

complex-valued functions on X. Every C∞(τ rs X) is a C∞(X)-module.

Let

P (T ∗X) =∞⊕

m=0

Pmhom(T ∗X)

be the graded algebra of smooth functions on the manifold T ∗X which are polynomialsin the argument ξ. Here Pm

hom(T ∗X) is the C∞(X)-module of homogeneous polynomialsof degree m in ξ. As usually, we denote by

Pm(T ∗X) =⊕

m′≤m

Pm′hom(T ∗X)

the module of polynomials of degree ≤ m. The module Pmhom(T ∗X) can be identified with

the module of symmetric contravariant tensor fields of rank m. In the domain of a localcoordinate system, a polynomial a ∈ Pm

hom(T ∗X) can be written down twofold:

a(x, ξ) =∑

|α|=m

aα(x)ξα = aj1...jm(x)ξj1 . . . ξjm , (2.2)

where aj1...jm(x) are symmetric in the indices j1, . . . , jm. Treating polynomials in coor-dinates, we will use either multi-index notations or tensor notations. Multi-indices will

5

be denoted by Greek letters while tensor indices, by Roman letters. For a multi-indexα = (α1, . . . , αn) and a sequence (j1, . . . , jm) with m = |α|, 1 ≤ ja ≤ n for 1 ≤ a ≤ m,we write α = 〈j1 . . . jm〉 if the sequence (j1, . . . , jm) coincides with the sequence

(1, . . . , 1︸ ︷︷ ︸α1

, 2, . . . , 2︸ ︷︷ ︸α2

, . . . , n, . . . , n︸ ︷︷ ︸αn

)

up to the order of elements. The coefficients of two sums in (2.2) are related by theequality

aα =m!

α!aj1...jm for α = 〈j1 . . . jm〉, m = |α|. (2.3)

Note that the multi-index α is in the low position in coefficients of (2.2) while the corre-sponding tensor indices j1 . . . jm are in the upper position. Probably, such disagreementof the two notation systems is uncomfortable for usage, but it reflects the matter of thefact that coefficients of a polynomial constitute a contravariant tensor field, i.e., the coeffi-cients of (2.2) are transformed by formula (2.1) with r = m and s = 0 under a coordinatechange.

Let

DO(X) =∞⋃

m=0

DOm(X)

be the algebra of linear differential operators with smooth coefficients on the manifoldX. Here DOm(X) is the C∞(X)-module of differential operators of order ≤ m. Anoperator A ∈ DOm(X), A : C∞(X) → C∞(X) can be represented in the domain of alocal coordinate system as follows:

A =∑

|α|≤m

aα(x)Dα, (2.4)

where D = −i∂x. Under a coordinate change, the coefficients aα of (2.4) are transformedby a rather cumbersome rule, i.e., aα do not constitute a tensor field. The situation canbe improved by using the covariant derivative.

We fix a symmetric connection∇ on the manifold X. For every r and s, the connectiondetermines the first order differential operator that is denoted by the same latter

∇ : C∞(τ rs X) → C∞(τ r

s+1X)

and is called the covariant derivative. In local coordinates this operator looks as follows:

∇luj1...jr

k1...ks=

∂

∂xluj1...jr

k1...ks+

r∑a=1

Γja

lpuj1...ja−1pja+1...jr

k1...ks−

s∑a=1

Γplka

uj1...jr

k1...ka−1pka+1...ks, (2.5)

where Γjkl are the Christoffel symbols of the connection ∇. The second order derivatives

satisfy the following commutator formula:

(∇l∇m −∇m∇l)uj1...jr

k1...ks=

r∑a=1

Rja

plmuj1...ja−1pja+1...jr

k1...ks−

s∑a=1

Rpkalmuj1...jr

k1...ka−1pka+1...ks, (2.6)

6

where (Rpjkl) is the curvature tensor of the connection ∇.

The following remark is important in order to avoid a possible misleading while usingthe covariant derivative. For given indices l, j1, . . . , jr, k1, . . . , ks, the derivative ∇lu

j1...jr

k1...ks

depends not only on uj1...jr

k1...ksbut on all components of the field u. Probably, the derivative

should be denoted more consistently by (∇u)j1...jr

lk1...ks. Nevertheless we will use the tradi-

tional notation (2.5). The same remark is valid for other differential operators on tensorfields.

We define the canonical isomorphism of the C∞(X)-modules

Pm(T ∗X) → DOm(X), a(x, ξ) 7→ a(x,−i∇) (2.7)

as follows. First of all, in the domain of a local coordinate system, we put for a multi-indexα = 〈j1 . . . jm〉

∇α = σ(j1 . . . jm) (∇j1 . . .∇jm) =1

m!

∑π∈Πm

∇jπ(1). . .∇jπ(m)

, (2.8)

where Πm is the group of all permutations of the set 1, . . . , m. So∇α is the symmetrizedcovariant derivative of order |α|. The family of ∇α with |α| = m constitutes a covariantobject, i.e., they are transformed under a coordinate change by the formula

∇′j1...jm

=∂xk1

∂x′j1. . .

∂xkm

∂x′jm∇k1...km , where ∇j1...jm = σ(j1 . . . jm) (∇j1 . . .∇jm) . (2.9)

As is seen from (2.5), the operators (−i∇)α and Dα coincide up to an operator of order|α| − 1. This means in particular that the family of ∇α with |α| ≤ m constitutes a basisfor the C∞(U)-module DOm(U) for the domain U of a local coordinate system. We defineisomorphism

Pm(T ∗U) → DOm(U)

by putting

a(x,−i∇) =∑

|α|≤m

aα(x)(−i∇)α for a(x, ξ) =∑

|α|≤m

aα(x)ξα. (2.10)

The isomorphism is independent of the choice of local coordinates as is seen from (2.3)and (2.9), and therefore it defines the global isomorphism (2.7). The polynomial a(x, ξ)is called the (full) geometric symbol of the differential operator a(x,−i∇) (with respectto a given connection). We will sometimes write a(x, ξ) = sym A if A = a(x,−i∇). Notethat the operator a(x,−i∇) is well defined on tensor fields of any rank

a(x,−i∇) : C∞(τ rs X) → C∞(τ r

s X).

In the present article we are interested, first of all, in investigating differential (and pseu-dodifferential) operators on scalar functions

a(x,−i∇) : C∞(X) → C∞(X).

7

In particular, the product of two operators is always understood in the latter sense.Nevertheless, sometimes we will need to apply a differential operator to tensor fields. See,for example, Corollary 2.3 below.

There is a danger of misleading while using higher order covariant derivatives in themulti-index form. For example, given a function u ∈ C∞(X) and multi-index α, we canconsider ∇αu as a function in the domain of a coordinate system and apply the operator∇β to the function. The result ∇β(∇αu) does not coincide with ∇β∇αu. Moreover,the family of functions ∇β(∇αu) does not constitute a tensor object while ∇β∇αu is acovariant tensor field of rank |α| + |β|. We will avoid using expressions like ∇β(∇αu) asfar as possible and will intend to write all our formulas in a most invariant form.

The remark of the previous paragraph is moreover valid for differential operatorswritten in the form a(x,−i∇) =

∑α aα(x)(−i∇)α. Given u ∈ C∞(X), a(x,−i∇)u is a

function and we can apply the operator ∇j = ∂/∂xj to the function

∇j(a(x,−i∇)u) =∑

α

(∂aα

∂xj(−i∇)αu + aα

∂(−i∇)αu

∂xj

).

Both summands on the right-hand side do not have a tensor character. Therefore we willmostly use the following form of the latter derivative:

∇j(a(x,−i∇)u) =∑

α

(∇jaα · (−i∇)αu + aα∇j(−i∇)αu

).

Here ∇jaα means the covariant derivative of the contravariant tensor field aα. So, we haveto treat the coefficients of a differential operator not as individual functions but rather ascomponents of a contravariant tensor field according to formula (2.3).

The covariant derivative is closely related to the exponential map of the connection∇.

Lemma 2.1 For u ∈ C∞(X),

∇αu(x) = ∂αv u(expxv)|v=0.

Proof. It suffices to prove the equality

∑

|α|=m

1

α!∇αu(x) · wα =

∑

|α|=m

1

α!∂α

v u(expxv)|v=0 · wα

for a vector w ∈ TxX. The latter equality can be rewritten in the form

(∇j1 . . .∇jmu(x)) wj1 . . . wjm =dm

dtmu(expxtw)|t=0.

Putting γ(t) = expx tw, we see that the latter equality is equivalent to the following one:

(∇j1 . . .∇jmu(x)) γj1(0) . . . γjm(0) =dm

dtmu(γ(t)|t=0.

The validity of the latter equality is evident because d/dt = γj(t)∇j.

8

Corollary 2.2 For every polynomial a ∈ P (T ∗X),

a(x,−i∇)u(x) = a(x, Dv)u(expxv)|v=0.

Corollary 2.3 Given two tensor fields u ∈ C∞(τ 0mX) and w ∈ C∞(τm

0 X), the Leibnitzformula

a(x,−i∇)(uj1...jmwj1...jm

)=

∑α

1

α!(

v

∇αa)(x,−i∇)uj1...jm · (−i∇)αwj1...jm

holds for any differential operator a(x,−i∇), where (v

∇αa)(x, ξ) = ∂αξ a(x, ξ).

We are going to derive the formula for the geometric symbol of the product of twodifferential operators. To this end, first of all, we introduce polynomials Rα,β(x, ξ) by theequalities

(−i∇)α(−i∇)β = Rα,β(x,−i∇). (2.11)

The leading term of the polynomial Rα,β(x, ξ) is ξα+β as is seen from (2.5). More precisely,Rα,β(x, ξ) = ξα+β + . . . , where dots mean a polynomial of degree ≤ |α|+ |β| − 2. Becauseof the crucial role of Rα,β, we will rewrite definition (2.11) in tensor notations. (2.11)means that for every function u ∈ C∞(X)

σ(j1 . . . jr)σ(k1 . . . ks)((−i∇)j1 . . . (−i∇)jr(−i∇)k1 . . . (−i∇)ksu

)

=r+s∑m=1

Rp1...pm

j1...jr,k1...ks(−i∇)p1 . . . (−i∇)pmu,

where (Rp1...pm

j1...jr,k1...ks) is a tensor field symmetric in each of three groups of its indices, and

R〈j1...jr〉,〈k1...ks〉(x, ξ) =r+s∑m=1

Rp1...pm

j1...jr,k1...ks(x)ξp1 . . . ξpm .

Hereafter σ(j1 . . . jr) is the symmetrization with respect to indices j1, . . . , jr defined in(2.8). The covariant character of Rα,β with respect to α and β is seen from the latterformula.

Several first polynomials Rα,β are as follows:

Rα,β = ξα+β for |α|+ |β| ≤ 2, (2.12)

Rα,0 = R0,α = ξα, (2.13)

R〈j〉,〈kl〉 = ξjξkξl − 1

3(Rp

klj + Rplkj)ξp, R〈jk〉,〈l〉 = ξjξkξl − 1

6(Rp

jlk + Rpklj)ξp, (2.14)

R〈jkl〉,〈m〉 = ξjξkξlξm +1

2σ(jkl)

(2Rp

jkmξl + (−i∇)jRpklm

)ξp, (2.15)

R〈j〉,〈klm〉 = ξjξkξlξm +1

2σ(klm)

(4Rp

kjlξm + (−i∇)kRpljm

)ξp, (2.16)

R〈jk〉,〈lm〉 = ξjξkξlξm

− 1

6σ(jk)σ(lm)

(8Rp

lmjξk + 4Rpjlkξm + 5(−i∇)jR

plmk + (−i∇)lR

pjmk

)ξp. (2.17)

9

In principle, any polynomial Rα,β can be written down explicitly in terms of the curva-ture tensor by repeatedly using the commutator formula (2.6); but the difficulty of thisprocedure grows fast with |α|+ |β|.

The product formula includes also some specific tensor operator that is called thehorizontal covariant derivative. We describe it briefly in the next three paragraphs. See[6] for a detailed presentation.

Recall that π : T ∗X → X is the cotangent bundle. The pull-back βrsX = π∗(τ r

s X)is a vector bundle over T ∗X which is called the bundle of semibasic tensors of rank(r, s), and sections of this bundle are called semibasic tensor fields on X. For such a fieldu ∈ C∞(βr

sX), the coordinates uj1...jr

k1...ks(x, ξ) are defined, in the domain of a local coordinate

system, which are transformed under a coordinate change by the same formula (2.1) as thecoordinates of an ordinary tensor field. From an analytical viewpoint, the only differencebetween ordinary tensor fields and semibasic tensor fields is that the coordinates of alatter field depend on (x, ξ). In particular, C∞(β0

0X) = C∞(T ∗X).If X is endowed with a connection, the horizontal covariant derivative

h

∇ : C∞(βrsX) → C∞(βr

s+1X)

is defined in local coordinates by the formula

h

∇luj1...jr

k1...ks=

∂

∂xluj1...jr

k1...ks+ Γq

lpξq∂

∂ξp

uj1...jr

k1...ks

+r∑

a=1

Γja

lpuj1...ja−1pja+1...jr

k1...ks−

s∑a=1

Γplka

uj1...jr

k1...ka−1pka+1...ks. (2.18)

In fact the result is independent of the choice of local coordinates. Comparing (2.5)and (2.18), we see that the latter formula has the additional term that accounts for the

dependence of u on ξ. In particular,h

∇u = ∇u if the coordinates of u are independent of ξ(sometimes such u are called basic tensor fields). The corresponding formula in [6] differsfrom (2.18) by the sign of the second term because [6] considers semibasic tensors overthe tangent bundle TX. The second order derivatives satisfy the commutator formula

[h

∇l,h

∇m]uj1...jr

k1...ks= Rq

plmξq∂

∂ξp

uj1...jr

k1...ks

+r∑

a=1

Rja

plmuj1...ja−1pja+1...jr

k1...ks−

s∑a=1

Rpkalmuj1...jr

k1...ka−1pka+1...ks,

which is also very similar to the corresponding formula (2.6).h

∇ commutes with the

vertical derivativev

∇ = ∂ξ : C∞(βrsX) → C∞(βr+1

s X).For the horizontal derivative, we will use the same notation as above:

h

∇α = σ(j1 . . . jm)

(h

∇j1 . . .h

∇jm

)for a multi-index α = 〈j1 . . . jm〉.

10

The semibasic covector field ξ ∈ C∞(β01X) is parallel with respect to

h

∇, i.e.,h

∇jξk = 0.This implies the following rule for differentiating polynomials:

h

∇β

(∑α

aα(x)ξα

)=

∑α

(∇βaα(x))ξα. (2.19)

For sufficiently close points x, y ∈ X, we denote by Ixy : TxX → TyX the parallel

transport of tangent vectors along the unique shortest geodesic from x to y. The corre-sponding parallel transport of covectors is t(Ix

y )−1 : T ∗xX → T ∗

y X. The following analog ofLemma 2.1 holds for the horizontal derivative:

h

∇αu(x, ξ) = ∂αv

(u(expxv, t(Ix

expxv)−1ξ)

)∣∣∣v=0

(2.20)

for u ∈ C∞(T ∗X) and for any multi-index α. We omit the proof of the formula which isquite similar to the proof of Lemma 2.1. Later we will prove a more general statement,see Lemma 7.5 below.

Formula (2.20) has the following corollary: for b ∈ C∞(T ∗X), the representation

∂αv

( v

∇βb(expxv, t(Ixexpxv)

−1ξ))∣∣∣

v=0=

∑γ≤α

∑

|δ|=|β|

h

∇γv

∇δb(x, ξ) · pαβγδ (x) (2.21)

holds for any multi-indices α and β, where pαβγδ (x) are some smooth functions in the

domain of a local coordinate system which are independent of b and ξ. Indeed, fix local

coordinates in a neighborhood of x and put u(x, ξ) =v

∇βb(x, ξ) for a multi-index β. Byinduction on |α|, one easily proves the representation

h

∇αu(x, ξ)− h

∇αv

∇βb(x, ξ) =∑γ<α

∑

|δ|=|β|

h

∇γv

∇δb(x, ξ) · pαβγδ (x)

which, together with (2.20), gives (2.21).

Theorem 2.4 The geometric symbol of the product

c(x,−i∇) = a(x,−i∇)b(x,−i∇)

of two differential operators is expressed through a(x, ξ) and b(x, ξ) by the formula

c(x, ξ) =∑

α

1

α!

v

∇αa∑

β,γ

1

γ!

(αβ

)(−i

h

∇)βv

∇γb · ρα−β,γ, (2.22)

where

(αβ

)= α!

β!(α−β)!are the binomial coefficients with

(αβ

)6= 0 only for β ≤ α; and

ρα,β(x, ξ) are polynomials expressed through polynomials (2.11) by the formula

ρα,β = (−1)|α|+|β|∑

λ,µ

(−1)|λ|+|µ|(

αλ

) (βµ

)ξα+β−λ−µRλ,µ. (2.23)

The degree of the polynomial ρα,β is ≤ |α|/4 + 3|β|/4.

11

Remark on notations. We use the central dot sometimes to fix the end of the action of

an operator. For instance, (−ih

∇)βv

∇γb·ρα−β,γ in (2.22) means the product of the derivative

(−ih

∇)βv

∇γb and of the function ρα−β,γ . The product can be written more conventionally in

the form ρα−β,γ(−ih

∇)βv

∇γb. We use the opposite order of factors taking in mind the futuregeneralization of Theorem 2.4 to the case of operators on vector bundles, see Theorems7.6 and 8.3 below. In the latter case ρα−β,γ is an operator-valued function and the orderof factors is essential.

Before proving the theorem, we make some remarks.Formula (2.22) can be rewritten in tensor notations:

c =∑m

1

m!

v

∇j1 . . .v

∇jma

m∑

l=0

∑p

1

p!

(ml

)(−i

h

∇)j1 . . . (−ih

∇)jl

v

∇k1 . . .v

∇kpb · ρjl+1...jm,k1...kp ,

where ρj1...jm,k1...kp = ρ〈j1...jm〉,〈k1...kp〉. Probably, the latter formula does not look so nice as(2.22) but sometimes it is more comfortable in usage.

ρ0,0 = 1 and ρα,β = 0 for |α| + |β| > 0 in the case of a flat connection when thecurvature tensor is equal to zero, and formula (2.19) coincides with the classical one:

c(x, ξ) =∑

α

1

α!∂α

ξ a ·Dαx b.

As follows from (2.12)–(2.17) and (2.23), several first polynomials ρα,β are

ρ0,0 = 1, ρα,0 = ρ0,α = 0 for |α| > 0, ρα,β = 0 for |α| = |β| = 1, (2.24)

ρ〈j〉,〈kl〉 = −1

3(Rp

klj + Rplkj)ξp, ρ〈jk〉,〈l〉 = −1

6(Rp

jlk + Rpklj)ξp, (2.25)

ρ〈j〉,〈klm〉 =1

2σ(klm)(−i∇)kR

pljmξp, ρ〈jkl〉,〈m〉 =

1

2σ(jkl)(−i∇)jR

pklmξp, (2.26)

ρ〈jk〉,〈lm〉 = −1

6σ(jk)σ(lm)

(5(−i∇)jR

plmk + (−i∇)lR

pjmk

)ξp. (2.27)

In view of (2.24) formula (2.22) can be rewritten in the form

c =∑

α

1

α!

v

∇αa · (−ih

∇)αb +∑

α

1

α!

v

∇αa∑

β<α

(αβ

) ∑γ>0

1

γ!(−i

h

∇)βv

∇γb · ρα−β,γ .

To prove the theorem, we need the following

Lemma 2.5 The linear system

∑

β

(αβ

)ξα−βxβ = bα (2.28)

has the unique solution given by the formula

xβ = (−1)|β|∑

γ

(−1)|γ|(

βγ

)ξβ−γbγ. (2.29)

12

Proof. Existence and uniqueness of the solution are evident because the systemhas a triangular matrix with units on the diagonal. To prove the second statement, wesubstitute (2.29) into (2.28) and after simple transformations arrive at the relation

∑

β

(−1)|β|(

αβ

)(βγ

)=

(−1)|α| for α = γ0 otherwise

which is evidently valid.Proof of Theorem 2.4. For a(x, ξ) =

∑λ

aλξλ, the geometric symbol of the operator

a(x,−i∇)(−i∇)µ is expressed by the formula

sym (a(x,−i∇)(−i∇)µ)(x, ξ) =∑

λ

aλ(x)Rλ,µ(x, ξ). (2.30)

Let us show that the latter formula can be rewritten in the form

sym (a(x,−i∇)(−i∇)µ)(x, ξ) =∑

α

1

α!

v

∇αa(x, ξ) · Sα,µ(x, ξ) (2.31)

with some polynomials Sα,µ. Indeed, comparing the two last formulas, we see that (2.31)is equivalent to the equality

∑α

1

α!

v

∇αa(x, ξ) · Sα,µ(x, ξ) =∑

λ

aλ(x)Rλ,µ(x, ξ).

Substituting the valuev

∇αa(x, ξ) =∑

λ

λ!

(λ− α)!aλ(x)ξλ−α

into the previous equality, we obtain

∑

α,λ

(λα

)aλ(x)Sα,µ(x, ξ)ξλ−α =

∑

λ

aλ(x)Rλ,µ(x, ξ).

Equating the coefficients of aλ on both sides of the latter equality, we arrive at the system

∑α

(λα

)ξλ−αSα,µ = Rλ,µ

in the sought polynomials Sα,µ. This is a system of type (2.28) and, by Lemma 2.5, thesolution is given by the formula

Sα,µ = (−1)|α|∑

λ

(−1)|λ|(

αλ

)ξα−λRλ,µ. (2.32)

Using the Leibnitz formula, we calculate

c(x,−i∇)u(x) = a(x,−i∇) (b(x,−i∇)u(x))

= a(x,−i∇)

(∑µ

bµ(x)(−i∇)µu(x)

)

=∑α,µ

1

α!(−i∇)αbµ(x) · v

∇αa(x,−i∇)(−i∇)µu(x).

13

We have thus proven that

c(x,−i∇) =∑α,µ

1

α!(−i∇)αbµ(x) · v

∇αa(x,−i∇)(−i∇)µ,

and therefore

c(x, ξ) =∑α,µ

1

α!(−i∇)αbµ(x) · sym

( v

∇αa(x,−i∇)(−i∇)µ)

.

Substituting the expression

sym( v

∇αa(x,−i∇) · (−i∇)µ)

=∑

β

1

β!

v

∇α+βa(x, ξ) · Sβ,µ(x, ξ),

which follows from (2.31), into the previous formula, we obtain

c(x, ξ) =∑

α,β,µ

1

α!β!(−i∇)αbµ(x) · v

∇α+βa(x, ξ) · Sβ,µ(x, ξ).

After evident changes of summation indices, this formula takes the form

c(x, ξ) =∑

α,β,µ

1

α!(α− β)!

v

∇αa(x, ξ) · Sα−β,µ(x, ξ)(−i∇)βbµ(x). (2.33)

We are going to show that (2.33) is equivalent to formula (2.22) with appropriatelychosen coefficients ρα,β. To this end we substitute the value

v

∇γb =∑

µ

µ!

(µ− γ)!bµξ

µ−γ

into (2.22) to obtain

c =∑

α,β,γ,µ

1

α!

(αβ

)(µγ

)v

∇αa · ρα−β,γ(−ih

∇)β(bµ(x)ξµ−γ

)

=∑

α,β,γ,µ

1

α!

(αβ

)(µγ

)v

∇αa · ρα−β,γ(−i∇)βbµ · ξµ−γ.

We have used (2.19) on the last step. Comparing the latter equality with (2.33), we arriveat the following system for the coefficients ρα,β:

∑γ

(µγ

)ξµ−γρα,γ = Sα,µ.

This is the system of the type (2.28) and, by Lemma 2.5, the solution is given by theformula

ρα,β = (−1)|γ|∑

µ

(−1)|µ|(

γµ

)ξγ−µSα,µ.

14

Substituting the value (2.32) for Sα,µ into the latter formula, we obtain (2.23).We have thus proven Theorem 2.4 with the exception of its last statement on the

degree of ρα,β. Most probably, it is the most difficult part of the theorem. It will beproven in Section 5.

We conclude the section with considering differential operators on half-densities. LetΩ1/2X be the one-dimensional vector bundle of half-densities over a manifold X. Thefiber over x ∈ X of the bundle is denoted by Ω1/2(TxX). Locally, in the domain U ⊂ Xof a coordinate system (x1, . . . , xn), a half-density u ∈ C∞(Ω1/2X) can be represented asu = λ(x)|dx|1/2 with a function λ ∈ C∞(U) which is transformed by the rule

λ(x) =√J λ′(x′), where J =

∣∣∣∣det∂x′

∂x

∣∣∣∣ , (2.34)

under a coordinate change. The function J satisfies the equality

J −1 ∂J∂xj

=∂xl

∂x′k∂2x′k

∂xj∂xl, (2.35)

see the arguments before formula (18.1.33) of [4].A differential operator A ∈ DOm(Ω1/2X) on the bundle Ω1/2(X) can be locally written

as follows:A(λ|dx|1/2) = a(x, D)λ(x) · |dx|1/2 (2.36)

with some polynomial a(x, ξ) =∑|α|≤m

aα(x)ξα. The polynomial is transformed by a

rather cumbersome rule under a coordinate change. In particular, the two leading termsa0(x, ξ) =

∑|α|=m

aα(x)ξα and a1(x, ξ) =∑

|α|=m−1

aα(x)ξα are transformed as follows:

a′0(x′, ξ′) = a0(x, ξ), (2.37)

a′1(x′, ξ′) = a1(x, ξ)− i

2

∂2x′p

∂xj∂xkξ′p

v

∇jv

∇ka0(x, ξ)− i

2J −1 ∂J

∂xk

v

∇ka0(x, ξ), (2.38)

where the same notations are used as in (1.2). Formulas (2.35) and (2.37)–(2.38) implythat the subprincipal symbol

a1s(x, ξ) = a1(x, ξ) +i

2

∂2a0(x, ξ)

∂xk∂ξk

(2.39)

is invariant, i.e., it is transformed by the formula

a′1s(x′, ξ′) = a1s(x, ξ) (2.40)

under a coordinate change.Now, let (X,∇) be a manifold with a fixed symmetric connection. The covariant

derivative∇ : C∞(Ω1/2X) → C∞(τ ∗X ⊗ Ω1/2X) (2.41)

15

on half-densities is defined in local coordinates by the formula

∇j(λ|dx|1/2) =

(∂λ

∂xj− 1

2Γjλ

)|dx|1/2,

whereΓj = Γp

jp,

and Γijk are the Christoffel symbols of the connection. One easily checks on the base of

(1.5) and (2.34) that the result is independent of the choice of a coordinate system.Operator (2.41) uniquely determines the covariant derivative

∇ : C∞(τ 0s X ⊗ Ω1/2X) → C∞(τ 0

s+1X ⊗ Ω1/2X) (2.42)

of tensor half-densities such that it is a derivative with respect to the tensor product. Thenwe define the symmetrized covariant derivative ∇αu of a half-density u ∈ C∞(Ω1/2X) by(2.8).

The isomorphism of C∞(X)-modules

Pm(T ∗X) → DOm(Ω1/2X), a(x, ξ) 7→ a(x,−i∇)

is defined by the same formula (2.10) as in the case of scalar functions. The isomorphismis independent of the choice of local coordinates. As above, we call a(x, ξ) the geometricsymbol of the differential operator A = a(x,−i∇), while the polynomial a(x, ξ) in (2.36)is called the coordinate-wise symbol of A. The leading terms of these polynomials arerelated by the formulas

a0(x, ξ) = a0(x, ξ), (2.43)

a1(x, ξ) = a1(x, ξ)− i

2Γp

jk(x)ξp

v

∇jv

∇ka0(x, ξ)− i

2Γj(x)

v

∇j a0(x, ξ). (2.44)

The subprincipal symbol is expressed through the geometric symbol as follows:

a1s(x, ξ) := a1s(x, ξ) = a1(x, ξ) +i

2

h

∇j

v

∇ja0(x, ξ). (2.45)

The parallel transport Ixy : TxX → TyX of tangent vectors induces the parallel trans-

port Jxy : Ω1/2(TxX) → Ω1/2(TyX) of half-densities. The following analog of Lemma

2.1∇αu(x) = ∂α

v

(Jexpxv

x u(expxv))∣∣∣

v=0(2.46)

holds for a half-density u ∈ C∞(Ω1/2X). Note that the expression in the parenthesesbelongs to the one-dimensional vector space Ω1/2(TxX), so the derivative on the right-hand side is well defined. The proof is omitted because it is similar to the proof of Lemma2.1. The corresponding analogs of Corollaries 2.2 and 2.3 hold for half-densities too.

16

3 Pseudodifferential operators

We fix a symmetric connection ∇ on a manifold X. The exponential map

expx : TxX ⊃ U → X (3.1)

of the connection is defined in some neighborhood U of the origin for every point x ∈ X.The connection ∇ will participate in our definition of a pseudodifferential operator onlyvia the exponential map. This means in particular that our assumption on symmetry of ∇does not restrict generality because the exponential map depends only on the symmetricpart of a connection.

For m ∈ R, the space of symbols Sm(T ∗X) of order ≤ m consists of all functionsa(x, ξ) ∈ C∞(T ∗X) satisfying the estimate

|∂αx ∂β

ξ a(x, ξ)| ≤ CK,α,β(1 + |ξ|)m−|β| for x ∈ K, ξ ∈ T ∗xX

in any local coordinate system for any multi-indices α, β and for any compact K ⊂X contained in the domain of the system. This space is defined invariantly, i.e., it isindependent of the choice of an atlas on X.

For a symbol a ∈ Sm(T ∗X), we say that a linear continuous operator

A = a(x,−i∇) : C∞0 (X) → D′(X) (3.2)

belongs to Ψm(X,∇) and has the geometric symbol a(x, ξ), if the Schwartz kernel of Ais C∞-smooth outside the diagonal ∆X = (x, x) ⊂ X ×X, and for every point x ∈ Xthere exists a neighborhood U of x such that

Au(x) = (2π)−n

∫

T ∗x X

∫

TxX

e−i〈v,ξ〉a(x, ξ)u(expxv) dvdξ (3.3)

for any function u ∈ C∞0 (U). Here 〈v, ξ〉 means the canonical pairing TxX × T ∗

xX → R,dv and dξ are dual densities on TxX and T ∗

xX respectively. Integral (3.3) is independentof the choice of such a pair (dv, dξ). The neighborhood U is assumed to be contained inthe range of expx, so the integrand is well defined.

One proves by standard arguments the following two facts: (1) every operator A ∈Ψm(X,∇) is continuous A : C∞

0 (X) → C∞(X) and A : E ′(X) → D′(X); (2) everyA ∈ Ψm(X,∇) can be represented as a sum A = A1 + A2 with a properly supportedA1 ∈ Ψm(X,∇) and smoothing A2 ∈ Ψ−∞(X).

In the case of an open set X ⊂ Rn and of the standard flat connection on Rn,expxv = x + v, and (3.3) coincides with the classical definition of a pseudodifferentialoperator:

Au(x) = (2π)−n

∫

Rn

∫

Rn

ei〈x−y,ξ〉a(x, ξ)u(y) dydξ. (3.4)

Let Ψm(X) = Ψm1,0(X) stand for the space of pseudodifferential operators of order ≤ m

according to the classical definition [4]. Up to smoothing operators, our definition givesthe same class of operators as the classical definition. More precisely, the equality

Ψm(X,∇)/Ψ−∞(X) = Ψm(X)/Ψ−∞(X) (3.5)

17

holds. Here we will prove the inclusion

Ψm(X,∇)/Ψ−∞(X) ⊂ Ψm(X)/Ψ−∞(X). (3.6)

The reverse inclusion will be proven in the next section. To prove (3.6) we note thatintegral (3.3) can be written after the change y = expxv of the integration variable asfollows:

Au(x) = (2π)−n

∫

T ∗x X

∫

U

ei〈exp−1x y,ξ〉a(x, ξ)u(y) dvx(y)dξ, (3.7)

where U is a neighborhood of x such that supp u ⊂ U and exp−1x y is defined for y ∈ U .

After introducing coordinates in U and T ∗xX, we can assume that U ⊂ Rn = T ∗

xX, and〈·, ·〉 is the standard dot-product in Rn. Then we can represent exp−1

x y as

exp−1x y = F (x, y)(y − x)

with a non-singular matrix F (x, y) depending smoothly on (x, y) ∈ U × U . After thechange of variables θ = tF (x, y)ξ, integral (3.7) takes the form

Au(x) = (2π)−n

∫

Rn

∫

Rn

ei〈x−y,θ〉a(x, y, θ)u(y) dydθ, (3.8)

where

a(x, y, θ) = a(x, (tF (x, y))−1θ)|det F (x, y)|−1 |dvx(y)||dy| ∈ Sm(U × U ×Rn).

As known [4], (3.8) is equivalent to the classical definition of Ψm(X) modulo Ψ−∞(X).Let us demonstrate that definition (3.3) covers the case of differential operators. In-

deed, if a(x, ξ) is a polynomial, we can transform integral (3.3) on using Corollary 2.2 asfollows:

Au(x) = (2π)−n

∫

T ∗x X

∫

TxX

(a(x,−Dv)e

−i〈v,ξ〉) u(expxv) dvdξ

= (2π)−n

∫

T ∗x X

∫

TxX

e−i〈v,ξ〉a(x,Dv)u(expxv) dvdξ

= a(x,Dv)u(expxv)|v=0 = a(x,−i∇)u(x).

This proves that A coincides with the differential operator a(x,−i∇) modulo Ψ−∞(X).The symbol a(x, ξ) can be recovered from the operator A = a(x,−i∇) by the formula

a(x, ξ) = Ay

(ψ(y)ei〈exp−1

x y,ξ〉)∣∣∣

y=x(mod S−∞(T ∗X)). (3.9)

Here ψ ∈ C∞0 (X) is a function supported in a small neighborhood of x and equal to

unity in a smaller neighborhood of x; and Ay means the operator A acting on y, i.e.,

18

x and ξ are considered as parameters on the right-hand side of (3.9). Indeed, puttingu(y) = ψ(y)ei〈exp−1

x y,ξ〉 in (3.3), we obtain

Ay

(ψ(y)e〈exp−1

x y,ξ〉)

= (2π)−n

∫

T ∗y X

∫

TyX

e−i〈v,η〉a(y, η)ψ(expy v)ei〈exp−1x expyv,ξ〉dvdη.

In particular

Ay

(ψ(y)e〈exp−1

x y,ξ〉)∣∣∣

y=x= (2π)−n

∫

T ∗x X

∫

TxX

ei〈v,ξ−η〉a(x, η) dvdη

+ (2π)−n

∫

T ∗x X

∫

TxX

ei〈v,ξ−η〉a(x, η)(ψ(expxv)− 1) dvdη.

The first integral on the right-hand side coincides with a(x, ξ) while the second integralbelongs to S−∞(T ∗X).

Together with (3.5), formula (3.9) establishes the canonical isomorphism

Ψm(X)/Ψ−∞(X) ∼= Sm(T ∗X)/S−∞(T ∗X) (3.10)

which is determined by the connection ∇. We will identify two operators whose differ-ence is a smoothing operator, and will do similar identification for symbols. So (3.10)can be read as follows: a connection determines a one-to-one correspondence betweenpseudodifferential operators and symbols.

Let us note some possibility that is not realized in the present article. While defininga pseudodifferential operator, we have assumed formula (3.3) to hold only for functionsu whose support is contained in a sufficiently small neighborhood of a point x. This is anecessary assumption because expxv is defined for small v in the case of a general connec-tion. If the exponential map (3.1) is defined on the whole of TxX for any x ∈ X (in sucha case ∇ is called a complete connection) and is a proper map, then formula (3.3) makessense for any u ∈ C∞

0 (X). In such a case one can define the space Ψmglobal(X,∇) by requir-

ing (3.3) to be valid for all u ∈ C∞0 (X). In the general case the analog of equality (3.5)

does not hold for Ψmglobal(X,∇), i.e., not every A ∈ Ψm

global(X,∇) is a pseudodifferentialoperator. The latter statement follows from the fact that the singular support of Au canbe more than singsupp u because of critical points of the exponential map. Neverthelessthe space Ψm

global(X,∇) is worth of studying since some important operators of this typearise naturally in integral geometry.

If we assume, moreover, the connection ∇ to be such that expx is a diffeomorphismof TxX onto X for any x, then Ψm

global(X,∇) ⊂ Ψm(X) and equality (3.5) holds withΨm

global(X,∇) in place of Ψm(X,∇). Such ∇′s can be called Hadamard connections byanalogy with Hadamard manifolds [1]. Some statements of the present article can bespecified in the case of Hadamard connections, but we will not do this.

We finish the section with the following statement whose proof is omitted because itis quite similar to the proof of Lemma 18.2.1 of [4]:

19

Lemma 3.1 Let b ∈ Sm(TX ⊕ T ∗X), and the operator A be defined by

Au(x) = (2π)−n

∫

T ∗x X

∫

TxX

e−i〈v,ξ〉b(x, v, ξ)u(expxv) dvdξ

for u ∈ C∞0 (X) supported in a sufficiently small neighborhood of x. If moreover the

Schwartz kernel of A is smooth outside the diagonal, then A ∈ Ψm(X,∇) with the geo-metric symbol

a(x, ξ) = (2π)−n

∫

T ∗x X

∫

TxX

ei〈v,ξ−η〉b(x, v, η) dvdη

which has the asymptotic expansion

a(x, ξ) ∼∑

j

1

j!〈∂v, Dξ〉jb(x, v, ξ)|v=0.

4 Expression for the geometric symbol in

local coordinates

Here we will show that every pseudodifferential operator A ∈ Ψm(X) can be uniquely, upto a smoothing operator, represented in the form (3.3); and will express the geometricsymbol a(x, ξ) through the coordinate-wise symbol a(x, ξ) of A. Since the classical defi-nition of a pseudodifferential operator is in fact a local one, we can assume without lossof generality X to be an open set of Rn endowed with a connection ∇.

By the classical definition of a pseudodifferential operator,

Au(x) = (2π)−n

∫

Rn

∫

Rn

ei〈x−y,η〉a(x, η)u(y) dydη (4.1)

for u ∈ C∞0 (X). Assume supp u ⊂ U ⊂ X and let U be so small that v = exp−1

x y isdefined for x, y ∈ U , and the representation expxv − x = F (x, v)v holds with a non-singular matrix F (x, v) depending smoothly on (x, v). Changing the integration variabley to v = exp−1

x y in (4.1), we obtain

Au(x) = (2π)−n

∫

Rn

∫

TxX

e−i〈v,tF (x,v)η〉a(x, η)u(expxv)

∣∣∣∣det∂ expxv

∂v

∣∣∣∣ dvdη

for x ∈ U . Then changing η to ξ = tF (x, v)η, we get

Au(x) = (2π)−n

∫

Rn

∫

TxX

e−i〈v,ξ〉c(x, v, ξ)u(expxv) dvdξ

with

c(x, v, ξ) = a(x, tF−1(x, v)ξ)|det F (x, v)|−1

∣∣∣∣det∂ expxv

∂v

∣∣∣∣ .

20

Finally, applying Lemma 3.1, we see that A ∈ Ψm(X,∇). We have thus proven (3.5).We proceed to computing the geometric symbol of an operator A ∈ Ψm(X) given by

(4.1) on an open set X ⊂ Rn endowed with a connection. In order to use formula (3.9),we choose a function ψ ∈ C∞

0 (X) with a small support which is unity in a neighborhoodof a given point x ∈ X. Putting u(y) = ψ(y)ei〈exp−1

x y,λξ〉 in (4.1), we obtain

Ay

(ψ(y)ei〈exp−1

x y,λξ〉)

= (2π)−n

∫ ∫ei(λ〈exp−1

x z,ξ〉−〈z−y,η〉)ψ(z)a(y, η)dzdη.

After the change z = z′ + y, η = λη′ the integral takes the form

Ay

(ψ(y)ei〈exp−1

x y,λξ〉)

=

(λ

2π

)n ∫ ∫eiλ(〈exp−1

x (y+z),ξ〉−〈z,η〉)ψ(y + z)a(y, λη)dzdη.

Putting y = x here and using (3.9), we obtain

a(x, λξ) =

(λ

2π

)n ∫ ∫eiλ(f(x,ξ;z)−〈z,η〉)ψ(x + z)a(x, λη)dzdη (4.2)

modulo S−∞(T ∗X), where

f(x, ξ; z) = 〈exp−1x (x + z), ξ〉. (4.3)

We are going to get an asymptotic expansion of integral (4.2) in λ on using a versionof the stationary phase method. Namely, applying Theorem 7.7.7 of [3] to integral (4.2),we obtain

a(x, λξ) ∼ eiλf(x,ξ;0)∑

j

1

j!〈Dz, ∂η/λ〉j

(eiλrx,ξ(z)a(x, λη)

)∣∣z=0,η=f ′z(x,ξ;0)

(4.4)

as λ →∞, where

rx,ξ(z) = f(x, ξ; z)− f(x, ξ; 0)− 〈f ′z(x, ξ; 0), z〉. (4.5)

By (4.3)f(x, ξ; 0) = 0, f ′z(x, ξ; 0) = ξ, rx,ξ(z) = 〈ρx(z), ξ〉

withρx(z) = exp−1

x (x + z)− z. (4.6)

Substituting these values into (4.4), we have

a(x, λξ) ∼∑

j

1

j!〈Dz, ∂η/λ〉j

(eiλ〈ρx(z),ξ〉a(x, λη)

)∣∣z=0,η=ξ

. (4.7)

We can eliminate the parameter λ from this relation on using the equality ∂η/λ = ∂λη.So, after the change ξ′ = λξ, η′ = λη, (4.7) takes the form

a(x, ξ) ∼∑

j

1

j!〈Dz, ∂η〉j

(ei〈ρx(z),ξ〉a(x, η)

)∣∣z=0,η=ξ

(4.8)

21

as |ξ| → ∞.In (4.8), the operator 〈Dz, ∂η〉j is applied to the product of two functions of which one

depends only on z and the other, only on η. Using this, we can rewrite (4.8) as follows:

a(x, ξ) ∼∑

α

1

α!Dα

z ei〈ρx(z),ξ〉∣∣∣∣∣z=0

· v

∇αa(x, ξ).

We have thus proven

Theorem 4.1 Let a pseudodifferential operator A of order ≤ m be given by formula(4.1) on an open set X ⊂ Rn endowed with a connection ∇. Then, being considered asan operator in Ψm(X,∇), A has the geometric symbol a(x, ξ) that is expressed througha(x, ξ) by the asymptotic series

a(x, ξ) ∼∑

α

1

α!ψα(x, ξ)

v

∇αa(x, ξ) (4.9)

with the coefficientsψα(x, ξ) = Dα

y ei〈ρ(x,y),ξ〉∣∣y=0

, (4.10)

whereρ(x, y) = exp−1

x (x + y)− y. (4.11)

Every function ψα(x, ξ) is a polynomial in ξ of degree ≤ |α|/2 with coefficients dependingsmoothly on x.

We have already proven this theorem except for the statement on the degree of ψα(x, ξ).The latter statement follows from the equalities

ρ(x, 0) = 0, ∂yρ(x, 0) = 0

by the same arguments as in the proof of Theorem 18.7.1 of [4].As is seen from (4.10), we need the Tailor series of the function ρ(x, y) in y to calculate

the coefficients ψα(x, ξ). The first terms of the series are as follows (compare with formula(17.15) of [2]):

(exp−1x (x + y)− y)i =

1

2Γi

jk(x)yjyk +1

6

(∂Γi

jk

∂xl+ Γi

jpΓpkl

)(x)yjykyl + . . . , (4.12)

where Γijk are the Christoffel symbols of the connection ∇. All coefficients ψα(x, ξ) can

be calculated, in principle, on the base of (4.10) and (4.12). Several first coefficients areas follows:

ψ0 = 1, ψ〈j〉 = 0, ψ〈jk〉 = −iΓpjkξp, (4.13)

ψ〈jkl〉 = −σ(jkl)

(∂Γp

jk

∂xl+ Γp

jqΓqkl

)ξp, (4.14)

ψ〈jklm〉 = −3σ(jklm)ΓpjkΓ

qlmξpξq + . . . , (4.15)

22

where the dots stand for a first order polynomial in ξ.Finally let us consider the case of a classical pseudodifferential operator A ∈ Ψm

cl (X).In this case the coordinate-wise symbol a(x, ξ) in (4.1) has the asymptotic expansion

a(x, ξ) ∼ a0(x, ξ) + a1(x, ξ) + a2(x, ξ) + . . . , (4.16)

where aj(x, ξ) is a homogeneous function of degree m − j in ξ for |ξ| ≥ 1. The similarexpansion

a(x, ξ) ∼ a0(x, ξ) + a1(x, ξ) + a2(x, ξ) + . . . (4.17)

holds for the geometric symbol. Starting from (4.9) and (4.13)–(4.15), one easily obtainsformulas (1.4) and (1.6)–(1.7) which connects the series (4.16) and (4.17). In principle,formulas like (1.6)–(1.7) can be obtained for every j, but the volume of calculations growsfast with j.

5 The symbol of the product of two operators

The last statement of Theorem 2.4 on the degrees of the polynomials ρα,β is not proved yet.Here we prove this statement and generalize Theorem 2.4 to the case of pseudodifferentialoperators.

Theorem 5.1 Let X be a manifold endowed with a connection ∇. Let one of the twooperators A = a(x,−i∇) ∈ Ψm1(X,∇) and B = b(x,−i∇) ∈ Ψm2(X,∇) be properlysupported. Then the product C = AB belongs to Ψm1+m2(X,∇), and the geometric symbolc(x, ξ) of C is expressed through a(x, ξ) and b(x, ξ) by the asymptotic series

c ∼∑

α

1

α!

v

∇αa ·∑

β,γ

1

γ!

(αβ

)(−i

h

∇)βv

∇γb · ρα−β,γ (5.1)

with the same coefficients ρα,β as in Theorem 2.4. A general term of the series belongs toSm1+m2−3|α|/4−|γ|/4(T ∗X); so the series converges asymptotically.

Proof. Take a point x0 ∈ X and a small neighborhood U of x0. The degree ofsmallness of U will be specified later. It is assumed in all formulas below that x ∈ U .Take u ∈ C∞

0 (U).The operator A is given by (3.3), and B is given by the same formula with b substituted

for a. Inserting one of these formulas into the other, we obtain

ABu(x) = (2π)−2n

∫

T ∗x

∫

Tx

∫

T ∗expxv

∫

Texpxv

e−i(〈v,ξ〉+〈w,η〉)a(x, ξ)b(expxv, η)u(expexpxv w) dwdηdvdξ.

(5.2)We sometimes use the abbreviations Tx = TxX and T ∗

x = T ∗xX. Since supp u ⊂ U , we

can assume without loss of generality that supp xa(x, ξ) ⊂ U and supp xb(x, ξ) ⊂ U .First of all we transform (5.2) to an integral over a fixed space independent of v. While

doing this, we should take care of preserving duality between w and η. Given x ∈ X andv ∈ TxX, we define the diffeomorphism

TexpxvX ⊃ W → Z ⊂ TxX, W 3 w 7→ z ∈ Z (5.3)

23

between small neighborhoods W and Z of the origins by the equation

expxz = expexpxv w. (5.4)

The neighborhood U of x0 is assumed to be so small that the range of expx contains Ufor every x ∈ U , and equation (5.4) has a unique small solution

z = z(x, v; w) (5.5)

for sufficiently small v and w. We define also the linear isomorphism

T ∗expxvX 3 η 7→ θ ∈ T ∗

xX (5.6)

by the formula

θ =t(∂z(x, v; w)

∂w

)−1

η. (5.7)

For fixed x and v, we have thus defined the change of variables

z = z(x, v; w)θ = θ(x, v; w, η)

(5.8)

with the unit Jacobian

det∂(z, θ)

∂(v, η)= 1. (5.9)

The inverse transform of (5.8) is as follows:

w(x, v; z) = exp−1expxv expx z

η(x, v; z, θ) =t(∂(exp−1

expxv expx z)∂z

)−1

θ(5.10)

We transform the two inner integrals of (5.2) according to the change of variables (5.8):

ABu(x) =

= (2π)−2n

∫

T ∗x

∫

Tx

∫

T ∗x

∫

Tx

e−i(〈v,ξ〉+〈w(x,v;z),η(x,v;z,θ)〉)a(x, ξ)b(expxv, η(x, v; z, θ))u(expxz) dzdθdvdξ.

This equality can be rewritten in the form

ABu(x) = (2π)−n

∫

T ∗x X

∫

TxX

e−i〈v,ξ〉d(x, v, ξ)u(expxv) dvdξ, (5.11)

where

d(x, v, ξ) = (2π)−n

∫

T ∗x X

∫

TxX

eiψ(x,v,ξ;w,η)a(x, η)b(expx w, tF (x, v, w)ξ) dwdη, (5.12)

24

F (x, v, w) =

(∂(exp−1

expxw expxv)

∂v

)−1

,

andψ(x, v, ξ; w, η) = 〈v, ξ〉 − 〈w, η〉 − 〈exp−1

expxw expxv, tF (x, v, w)ξ〉.From (5.11) we conclude with the help of Lemma 3.1 that the geometric symbol c(x, ξ)

of the operator C = AB is

c(x, ξ) = (2π)−n

∫

TxX

∫

T ∗x X

ei〈v,ξ−η〉d(x, v, η)dηdv.

Inserting expression (5.12) for d(x, v, ξ) in the latter formula, we obtain

c(x, ξ) = (2π)−2n

∫ ∫ ∫ ∫eiχ(x,ξ;v,η,w,ζ)a(x, ζ)b(expx w, tF (x, v, w)η) dwdζdηdv, (5.13)

whereχ(x, ξ; v, η, w, ζ) = 〈v, ξ〉 − 〈w, ζ〉 − 〈exp−1

expxw expxv, tF (x, v, w)η〉. (5.14)

In (5.13), the integrals with respect to v and w are taken over TxX and with respect toη and ζ, over T ∗

xX.We are going to investigate the asymptotics of (5.13) as |ξ| → ∞. First of all we

distinguish explicitly the asymptotic parameter λ. To this end we change variables in(5.13) as follows:

ξ = λξ′, η = λζ ′, ζ = λη′, v = v′ + w′, w = v′.

After the change, integral (5.13) takes the form

c(x, λξ) =

(λ

2π

)2n∫ ∫ ∫ ∫eiλϕ(x,ξ;v,η,w,ζ)a(x, λη)b(expxv, λtH(x, v, w)ζ)dvdηdwdζ

(5.15)with the phase function

ϕ(x, ξ; v, η, w, ζ) = 〈v + w, ξ〉 − 〈v, η〉 − 〈f(x, v; w), ζ〉, (5.16)

where

H(x, v, w) = F (x,w, v + w) =

(∂(exp−1

expxv(expx(v + w)))

∂w

)−1

(5.17)

andf(x, v; w) = H(x, v, w) · exp−1

expxv(expx(v + w)). (5.18)

We consider ϕ(x, ξ; v, η, w, ζ) as a function of the integration variables (v, η, w, ζ) while xand ξ are considered as fixed parameters.

25

Lemma 5.2 Considered as a function of four arguments (v, η, w, ζ), the phase function(5.16) has the unique critical point

v = w = 0, η = ζ = ξ, (5.19)

and the Hessian at this point is

ϕ′′vv ϕ′′vη ϕ′′vw ϕ′′vζ

ϕ′′ηv ϕ′′ηη ϕ′′ηw ϕ′′ηζ

ϕ′′wv ϕ′′wη ϕ′′ww ϕ′′wζ

ϕ′′ζv ϕ′′ζη ϕ′′ζw ϕ′′ζζ

=

0 −I 0 0−I 0 0 00 0 0 −I0 0 −I 0

where I is the identity matrix.

Proof. Differentiating (5.16), we have

ϕ′v = ξ − η − 〈f ′v, ζ〉; ϕ′η = −v; ϕ′w = ξ − 〈f ′w, ζ〉; ϕ′ζ = −f. (5.20)

So, v = f = 0 at a critical point and consequently

exp−1expxv(expx(v + w)) = w.

Now, (5.16) and (5.18) imply that 0 = f = w at the critical point. (5.18) implies alsothat f(x, 0; w) = w and consequently

f ′w(x, 0; 0) = I, f ′′ww(x, 0; 0) = 0. (5.21)

The third equality in (5.20) together with (5.21) gives us that ζ = ξ at the critical point.So, it remains to prove that η = ξ at the critical point. By the first equality in (5.20),this is equivalent to f ′v(x, 0; 0) = 0.

Definition (5.18) for f can be rewritten in the form

f(x, v; w) =

(∂z(x, v; w)

∂w

)−1

z(x, v; w), (5.22)

where z(x, v; ·), for fixed x and v, is the map

z(x, v; ·) : TxX → TexpxvX, z(x, v; w) = exp−1expxv(expx(v + w)). (5.23)

By (5.22), z and f are connected by the equation

z′w(x, v; w) · f(x, v; w) = z(x, v; w). (5.24)

We are looking for the derivative f ′v. To this end we differentiate (5.24) with respect to v:

z′wf ′v + z′′vwf = z′v. (5.25)

As we know, f = 0 at the critical point, and the latter equality takes the form

z′wf ′v = z′v. (5.26)

26

As is seen from (5.23), z(x, 0; w) = w and consequently z′w(x, 0; w) = I. Now, (5.26) gives

f ′v(x, 0; 0) = z′v(x, 0; 0). (5.27)

The question is thus reduced to finding the derivative z′v(x, 0; 0).By (5.23), the map z(x, v; ·) is defined by the equation

expexpxv z(x, v; w) = expx(v + w). (5.28)

For y in a neighborhood of x and for sufficiently small h ∈ TyX, we have in local coordi-nates

(expy h)i = yi + hi − 1

2Γi

jk(y)hjhk + o(|h|2). (5.29)

In particular,

(expxv)i = xi + vi − 1

2Γi

jk(x)vjvk + o(|v|2), (5.30)

(expx(v + w))i = xi + vi + wi − 1

2Γi

jk(x)(vj + wj)(vk + wk) + o(|v|2 + |w|2). (5.31)

Putting y = expxv and h = z = z(x, v; w) in (5.29), we obtain

(expexpxv z(x, v; w)

)i= (expxv)i + zi − 1

2Γi

jk(expxv)zjzk + o(|z|2).

Inserting expression (5.30) for expxv in the latter formula, we get

(expexpxv z(x, v; w)

)i= xi +vi +zi− 1

2Γi

jk(x)vjvk− 1

2Γi

jk(x+v+o(|v|))zjzk +o(|v|2 + |z|2).

Substituting the last expression and (5.31) into (5.28), one easily obtains

z(x, v; w) = w + o(|v|+ |w|).Consequently

z′v(x, 0; 0) = 0, z′w(x, 0; 0) = I. (5.32)

Now, from (5.20), (5.27) and (5.32), we see that ξ = η at the critical point. We have thusproven that the function ϕ has the unique critical point (v, η, w, ζ) = (0, ξ, 0, ξ).

We proceed to calculating the Hessian of ϕ at the critical point. To this end we first ofall will prove that the function f(x, v; w) has the zero Hessian with respect to (v, w) in thepoint (v, w) = (0, 0). Indeed, we have already proven (see (5.21)) that f ′′ww(x, 0; 0) = 0.In order to find other second order derivatives f ′′vv and f ′′vw, we differentiate (5.24) twice

z′wf ′′vv + 2z′′vwf ′v + z′′′vvwf = z′′vv,

z′wf ′′vw + z′′wwf ′v + z′′vwf ′w + z′′′vwwf = z′′vw,

z′wf ′′ww + 2z′′wwf ′v + z′′′wwwf = z′′ww.

Since f = 0, f ′v = 0, f ′w = I, z′w = I, z′′ww = 0 at the point (v, w) = (0, 0), the previousformulas are simplified to the following:

f ′′vv = z′′vv, (5.33)

27

f ′′vw + z′′vw = z′′vw, i.e., f ′′vw = 0,

f ′′ww + z′′ww = z′′ww, i.e., f ′′ww = 0.

We have thus proven that f ′′vw(x, 0; 0) = f ′′ww(x, 0; 0) = 0. To prove the equality f ′′vv(x, 0; 0) =0, it suffices, by (5.33), to demonstrate that z′′vv(x, 0; 0) = 0.

We choose a geodesic at x coordinate system in a neighborhood of x. In these coordi-nates

x = 0, Γijk(x) = 0, expxv = v.

As is seen from (5.28), the function z(x, ·; ·) is determined by the equation

expv z(x, v; w) = v + w or z(x, v; w) = exp−1v (v + w)

in these coordinates. Expanding exp−1v into the Tailor series, we have

zi(x, v; w) = (vi + wi)− vi +1

2Γi

jk(v)wjwk + o(|v|2 + |w|2).

This implies that all second order derivatives of the functions zi(x, v; w) with respect to(v, w) vanish at the point (0, 0).

We have thus proven that the Hessian of f vanishes at the critical point. On usingthis fact, the last statement of Lemma 5.2 is proved by differentiating (5.16) and takingthe equalities f ′v(x, 0; 0) = 0 and f ′v(x, 0; 0) = I into account.

We will apply the stationary phase method in the following form (see Theorems 7.7.5and 7.7.6 of [3]): ∫

RN

eiλϕ(z)u(z)dz ∼

∼ eiλϕ(z0)e−iπσ(Q)/4|det Q|−1/2

(2π

λ

)N2 ∑

j

λ−j

j!〈−1

2Q−1D, ∂〉j (

eλg(z)u(z))∣∣∣∣

z=z0

,

where z0 is a unique nondegenerate critical point of ϕ(z), Q = ϕ′′(z0), σ(Q) is the signatureof Q, and

g(z) = ϕ(z)− ϕ(z0)− 1

2〈ϕ′′(z0)(z − z0), z − z0〉.

In the case of integral (5.15), by Lemma 5.2, we have

N = 4n, z = (v, η, w, ζ), z0 = (0, ξ, 0, ξ),

det Q = 1, σ(Q) = 0, 〈−1

2Q−1D, ∂〉 = Dv∂η + Dw∂ζ ,

andg(x; v, η, w, ζ) = 〈w − f(x, v; w), ζ〉,

where f(x, v; w) is defined by (5.18). Note that g(x; v, η, w, ζ) is independent of η anddepends linearly on ζ.

28

Applying the stationary phase method to integral (5.15), we obtain

c(x, λξ) ∼∑

j

λ−j

j!(Dv∂η + Dw∂ζ)

j(eiλ〈h(x;v,w),ζ〉a(x, λη)b(expxv, λtH(x; v, w)ζ)

)∣∣v=w=0η=ζ=ξ

,

(5.34)where

h(x; v, w) = w − f(x; v, w). (5.35)

First of all we will eliminate the parameter λ from (5.34). To this end we notice thatλ−j(Dv∂η + Dw∂ζ)

j = (Dv∂λη + Dw∂λζ)j. Therefore (5.34) can be rewritten as follows:

c(x, ξ) ∼∑

j

1

j!(Dv∂η + Dw∂ζ)

j(ei〈h(x;v,w),ζ〉a(x, η)b(expxv, tH(x; v, w)ζ)

)∣∣v=w=0η=ζ=ξ

.

(5.36)The asymptotic expansion is understood to be with respect to |ξ| → ∞.

Further transformations of the series (5.36) are based on the following

Lemma 5.3 The functions h(x; v, w) and H(x; v, w) in (5.36) satisfy the conditions

h(x; v, 0) = 0, h′w(x; v, 0) = 0, H(x; v, 0) = (exp′x(v))−1, (5.37)

where exp′x(v) : TxX → TexpxvX is the differential of the map expx at v.

Proof. By definitions (5.17), (5.22) and (5.35),

h(v, w) = w − f(v, w), H(v, w) =

(∂z(v, w)

∂w

)−1

, f(v, w) =

(∂z

∂w

)−1

z(v, w), (5.38)

wherez(v, w) = exp−1

expxv(expx(v + w)). (5.39)

We do not write the argument x because it is fixed. As is seen from (5.38), the statementof the lemma follows from the equalities

z(v, 0) = 0, (5.40)

z′w(v, 0) = exp′x(v), (5.41)

f ′w(v, 0) = I. (5.42)

Equality (5.40) follows immediately from (5.39).Let us prove (5.41). As we have shown above, the function z(v, w) is defined by the

equationexpv z(v, w) = v + w (5.43)

in the geodesic coordinates at x, and the operator exp′x(v) has the identity matrix in thesecoordinates. If we substitute the expression

expv z(v, w) = v + z(v, w) + O(|z|2)

29

into (5.43), then it becomes clear that

z(v, w) = w + O(|w|2).

Therefore the operator z′w(v, 0) has the identity matrix.Let us prove (5.42). Differentiating (5.24), we have

z′wf ′w + z′′wwf = z′w.

Since f |w=0 = 0 z′w|w=0 = exp′x(v) is nondegenerate, the latter equality implies (5.42).The lemma is proved.

We transform series (5.36) as follows.First, substituting the expression

1

j!(Dv∂η + Dw∂ζ)

j =∑

|α|+|β|=j

1

α!β!∂α

η Dαv ∂β

ζ Dβw

into (5.36) and using the fact that a(x, η) is independent of (v, w, ζ), we obtain

c(x, ξ) ∼∑

α

1

α!

v

∇αa(x, ξ) ·Dαv

[∑

β

1

β!∂β

ξ Dβw

(ei〈h(x;v,w),ξ〉b(expxv, tH(x; v, w)ξ)

)∣∣w=0

]

v=0

.

(5.44)By Lemma 5.3, the function 〈h(x; v, w), ξ〉 of the variable w vanishes at w = 0 together

with the first order derivatives. This implies that

Dβw ei〈h(x;v,w),ξ〉∣∣

w=0= qβ(x, v, ξ), (5.45)

where qβ(x, v, ξ) are some polynomials of degree ≤ |β|/2 in ξ with coefficients dependingsmoothly on (x, v) (Compare with the arguments in the proof of Theorem 18.1.17 of [4]).

We will now validate the representation

Dβw

(ei〈h(x;v,w),ξ〉b(expxv, tH(x; v, w)ξ)

)∣∣w=0

=∑

|γ|≤|β|

v

∇γb(expxv, t(exp′x(v))−1ξ) · qβγ (x, v, ξ)

(5.46)with some polynomials qβ

γ (x, v, ξ) of degree ≤ |β|/2 + |γ| in ξ whose coefficients dependsmoothly on (x, v). Indeed, by (5.45) and (5.37),

Dβw

(ei〈h(x;v,w),ξ〉b(expxv, tH(x; v, w)ξ)

)∣∣w=0

=∑

λ+µ=β

(βλ

)Dλ

wb(expxv, tH(x; v, w)ξ)∣∣w=0

·Dµw ei〈h(x;v,w),ξ〉∣∣

w=0

=∑

λ+µ=β

∑

|γ|≤|λ|

(βλ

)v

∇γb(expxv, t(exp′x(v))−1ξ)qλ,γ(x, v, ξ)qµ(x, v, ξ),

30

where qλ,γ(x, v, ξ) are some polynomials of degree ≤ |γ| in ξ. The latter formula can berewritten as follows:

Dβw

(ei〈h(x;v,w),ξ〉b(expxv, tH(x; v, w)ξ)

)∣∣w=0

=∑

|γ|≤|β|

v

∇γb(expxv, t(exp′x(v))−1ξ)

( ∑

λ+µ=β

(βλ

)qλ,γ(x, v, ξ)qµ(x, v, ξ)

).

The degree of the polynomial in the parentheses is ≤ |µ|/2 + |γ| ≤ |β|/2 + |γ|. We havethus proven (5.46).

Let us now validate the representation

∂βξ Dβ

w

(ei〈h(x;v,w),ξ〉b(expxv, tH(x; v, w)ξ)

)∣∣w=0

=∑

|β|/2≤|γ|≤2|β|

v

∇γb(expxv, t(exp′x(v))−1ξ) · pβγ(x, v, ξ), (5.47)

where pβγ(x, v, ξ) are some polynomials of degree ≤ |γ|− |β|/2 in ξ. Indeed, differentiating

(5.46), we have

∂βξ Dβ

w

(ei〈h(x;v,w),ξ〉b(expxv, tH(x; v, w)ξ)

)∣∣w=0

=∑

|γ|≤|β|

∑

λ+µ=β

(βλ

)v

∇γ+λb(expxv, t(exp′x(v))−1ξ) · rλ(x, v)∂µξ qβ

γ (x, v, ξ)

=∑

|γ|≤2|β|

v

∇γb(expxv, t(exp′x(v))−1ξ)

( ∑

λ+µ=β

(βλ

)rλ(x, v)∂µ

ξ qβγ−λ(x, v, ξ)

).

The degree of the polynomial in the parentheses is not more than

|β|/2 + |γ − λ| − |µ| = |β|/2 + |γ| − |λ + µ| = |γ| − |β|/2.This proves (5.47).

Summing (5.47) over β, we get

∑

β

1

β!∂β

ξ Dβw

(ei〈h(x;v,w),ξ〉b(expxv, tH(x; v, w)ξ)

)∣∣w=0

=∑

γ

v

∇γb(expxv, t(exp′x(v))−1ξ)

∑

|β|≥|γ|/2

1

β!pβ

γ(x, v, ξ)

.

The polynomial pβγ is not zero only if |γ| − |β|/2 ≥ 0. Therefore the inner sum on the

right-hand side is finite, and the latter formula can be rewritten as follows:

∑

β

1

β!∂β

ξ Dβw

(ei〈h(x;v,w),ξ〉b(expxv, tH(x; v, w)ξ)

)∣∣w=0

=∑

γ

v

∇γb(expxv, t(exp′x(v))−1ξ) · qγ(x, v, ξ), (5.48)

31

where qγ(x, v, ξ) are some polynomials in ξ whose coefficients depend smoothly on (x, v).The degree of qγ is not more than

max|γ|/2≤|β|≤2|γ|

deg pβγ ≤ max

|γ|/2≤|β|≤2|γ|(|γ| − |β|/2) ≤ 3

4|γ|.

Differentiating (5.48) with respect to v, we obtain

Dαv

[∑

β

1

β!∂β

ξ Dβw

(ei〈h(x;v,w),ξ〉b(expxv, tH(x; v, w)ξ)

)]

v=w=0

=∑

β≤α

∑γ

Dβv

( v

∇γb(expxv, t(exp′x(v))−1ξ))∣∣∣

v=0· qα

βγ(x, ξ), (5.49)

where qαβγ(x, ξ) are some polynomials in ξ whose coefficients depend smoothly on x and

deg qαβγ ≤ 3|γ|/4. (5.50)

Let validate the equality

Dβv

( v

∇γb(expxv, t(exp′x(v))−1ξ))∣∣∣

v=0=

∑

λ≤β

∑

|γ|≤|µ|≤|β−λ+γ|(−i

h

∇)λv

∇µb(x, ξ) · pβγλµ(x, ξ),

(5.51)where pβ

λµ(x, ξ) are some polynomials in ξ with coefficients depending smoothly on x andwith

deg pβγλµ ≤ |µ| − |γ|. (5.52)

To this end, we represent the operator t(exp′x(v))−1 : T ∗xX → T ∗

expxvX as the productt(exp′x(v))−1 = t(Ix

expxv)−1L(x, v), where

L(x, v) = t(Ixexpxv) · t(exp′x(v))−1 : T ∗

xX → T ∗xX.

On using the representation, we can write

v

∇γb(expxv, t(exp′x(v))−1ξ) =v

∇γb(expxv, t(Ixexpxv)

−1η)∣∣∣η=L(x,v)ξ

.

Applying the operator Dβv to the latter equality and differentiating the right-hand side as

a composite function, we get the formula

Dβv

( v

∇γb(expxv, t(exp′x(v))−1ξ))

=∑

λ+µ≤β

DλvDµ

η

( v

∇γb(expxv, t(Ixexpxv)

−1η))∣∣∣

η=L(x,v)ξ· pβ

λµ(x, v, ξ) (5.53)

with some coefficients pβλµ(x, v, ξ) that are expressed polynomially through the function

η(x, v, ξ) = L(x, v)ξ and its derivatives with respect to v. Since η(x, v, ξ) depends linearlyon ξ, pβ

λµ(x, v, ξ) is a polynomial in ξ. Moreover, the degree of the polynomial is ≤ |µ|

32

because the appearance of any new component of the covector ξ in pβλµ(x, v, ξ) relates to

applying the operator Dηjto the first factor on the right-hand side of (5.53). In view of

the evident equality

Dµη

( v

∇γb(expxv, t(Ixexpxv)

−1η))

=v

∇γ+µb(expxv, t(Ixexpxv)

−1η) · rµ(x, v),

formula (5.53) can be rewritten as follows:

Dβv

( v

∇γb(expxv, t(exp′x(v))−1ξ))

=∑

λ+µ≤β

Dλv

( v

∇γ+µb(expxv, t(Ixexpxv)

−1η))∣∣∣

η=L(x,v)ξ· pβ

λµ(x, v, ξ) (5.54)

with some new polynomial coefficients pβλµ(x, v, ξ) satisfying the same estimate deg pβ

λµ ≤|µ|. Putting v = 0 in (5.54), we get

Dβv

( v

∇γb(expxv, t(exp′x(v))−1ξ))∣∣∣

v=0

=∑

λ+µ≤β

Dλv

( v

∇γ+µb(expxv, t(Ixexpxv)

−1ξ))∣∣∣

v=0· pβ

λµ(x, ξ).

Replacing the summation index µ with ν = γ + µ, we rewrite the latter equality in theform

Dβv

( v

∇γb(expxv, t(exp′x(v))−1ξ))∣∣∣

v=0

=∑

λ≤β

∑

γ≤ν≤β+γ−λ

Dλv

( v

∇νb(expxv, t(Ixexpxv)

−1ξ))∣∣∣

v=0· pβ

λν(x, ξ), (5.55)

wheredeg pβ

λν ≤ |ν − γ|. (5.56)

By (2.21),

Dλv

( v

∇νb(expxv, t(Ixexpxv)

−1ξ))∣∣∣

v=0=

∑

δ≤λ

∑

|µ|=|ν|(−i

h

∇)δv

∇µb(x, ξ) · pλνδµ(x).

Substituting this value into (5.55), we obtain (5.51)–(5.52).Substituting expression (5.51) into (5.49), we obtain

Dαv

[∑

β

1

β!∂β

ξ Dβw

(ei〈h(x;v,w),ξ〉b(expxv, tH(x; v, w)ξ)

)]

v=w=0

=∑

β≤α

∑γ

∑

λ≤|β|

∑

|γ|≤|µ|≤|β−λ+γ|(−i

h

∇)λv

∇µb · pβγλµq

αβγ. (5.57)

33

By (5.50) and (5.52), the degree of the polynomial pβγλµq

αβγ is not more than

|µ| − |γ|+ 3

4|γ| = 3

4|µ|+ 1

4(|µ| − |γ|) ≤ 3

4|µ|+ 1

4(|β| − |λ|) ≤ 1

4(|α| − |λ|) +

3

4|µ|.

Therefore equality (5.57) can be rewritten in the form

Dαv

[∑

β

1

β!∂β

ξ Dβw

(ei〈h(x;v,w),ξ〉b(expxv, tH(x; v, w)ξ)

)]

v=w=0

=∑

β≤α

∑γ

(−ih

∇)βv

∇γb(x, ξ) · pαβγ(x, ξ), (5.58)

where pαβγ(x, ξ) are some polynomials of degree ≤ (|α| − |β|)/4 + 3|γ|/4 in ξ.

Substituting (5.58) into (5.44), we obtain

c(x, ξ) ∼∑

α

1

α!

v

∇αa∑

β≤α

∑γ

(−ih

∇)βv

∇γb · pαβγ, (5.59)

where

deg pαβγ ≤

1

4(|α| − |β|) +

3

4|γ|. (5.60)

By (5.60), a general term of series (5.59) belongs to Sm1+m2−3|α|/4−|γ|/4(T ∗X). Thereforethe series converges asymptotically.

Formula (5.59) is proved for any two pseudodifferential operators A = a(x,−i∇) andB = b(x,−i∇). In particular, it holds for differential operators. Comparing (5.59) with

Theorem 2.4, we see that pαβγ must coincide with

(αβ

)ρα−β,γ/γ!. This proves Theorem

5.1 as well as the last statement of Theorem 2.4 on the degree of ρα,β.Remark. The statement

deg ρα,β ≤ 1

4|α|+ 3

4|β|

of Theorem 2.4 can be improved to the estimate

deg ρα,β ≤ 1

8|α|+ 3

4|β| (5.61)

by the following argument. One can easily show that the function L(x, v) in (5.53) hasthe following property: the first order derivatives of L(x, v) with respect to v vanish atv = 0. On using this fact, one proves that the coefficient pβ

λµ in (5.55) can be nonzero onlyfor |µ| ≤ |β − λ|/2. On using the latter inequality and repeating the rest of arguments inthe proof, we obtain (5.61).

34

6 Dual operator

Discussing the dual operator, we should consider operators on half-densities because thenatural L2-product

(u,w)L2 =

∫

X

uw

is defined for u,w ∈ C∞0 (Ω1/2X).

Let (X,∇) be a manifold with a connection, and a ∈ Sm(T ∗X). We say that a linearcontinuous operator

A = a(x,−i∇) : C∞0 (Ω1/2X) → D′(Ω1/2X)

belongs to Ψm(Ω1/2X,∇) and has the geometric symbol a(x, ξ) if the Schwartz kernel of Ais C∞-smooth outside the diagonal and, for every point x ∈ X, there exists a neighborhoodU of x such that

Au(x) = (2π)−n

∫

T ∗x X

∫

TxX

e−i〈v,ξ〉a(x, ξ)Jexpxvx u(expxv) dvdξ (6.1)

for any half-density u ∈ C∞(Ω1/2X) with supp u ⊂ U . Here Jexpxvx : Ω1/2(TexpxvX) →

Ω1/2(TxX) is the parallel transport of half-densities along the geodesic t 7→ expx tv, andthe other notations are the same as in (3.3). The integrand of (6.1) belongs to the one-dimensional vector space Ω1/2(TxX) for all v ∈ TxX and ξ ∈ T ∗

xX, so the integral is welldefined.

If a(x, ξ) is a polynomial in ξ, then (6.1) coincides with the differential operatora(x,−i∇) defined at the end of Section 2. This can be easily proved with the help of(2.46).

Theorem 6.1 For A ∈ Ψm(Ω1/2X,∇), the dual operator A∗ belongs also to Ψm(Ω1/2X,∇)and has the geometric symbol b(x, ξ) which is expressed through the geometric symbola(x, ξ) of A by the asymptotic series

b(x, ξ) ∼∑

α

1

α!(−i

h

∇)αv

∇αa(x, ξ).

To prove the theorem, we need the following two lemmas. The proof of the first lemmais omitted since it follows immediately from definitions.

Lemma 6.2 Let x, y ∈ X be two sufficiently close points belonging to the domain of a localcoordinate system (x1, . . . , xn). Let Ix

y : TxX → TyX be the parallel transport of tangentvectors along the shortest geodesic joining x and y, and Jy

x : Ω1/2(TyX) → Ω1/2(TxX) bethe corresponding parallel transport of half-densities. For a half-density u = λ(x)|dx|1/2

the equality(Jy

xu)(x) = λ(y)|det Ixy |1/2|dx|1/2

holds.

35

Lemma 6.3 (Reciprocity relation) Let two points x, y ∈ X belong to the domain of alocal coordinate system, and γ : [0, 1] → X be a geodesic from x to y, i.e., γ(0) = x andγ(1) = y. Put v = γ(0) ∈ TxX and w = −γ(1) ∈ TyX. Let Ix

y = (Iyx)−1 : TxX → TyX be

the parallel transport along γ. Then

|det exp′x(v)|/|det Ixy | = |det exp′y(w)|/|det Iy

x |,where exp′x(v) is the differential of expx at v. All determinants are calculated in the samecoordinate system.

Proof. First of all we note that the both sides of the equality are independent ofthe choice of coordinates. Therefore it suffices to prove the equality for some coordinatesystem.

Let γ(t) = e1(t), e2(t), . . . , en(t) be a parallel basis along γ. Choose Fermi’s coor-dinates corresponding to the basis. This means that a point p has the coordinates(t = x1, x2, . . . , xn) if p = expγ(t)

∑ni=2 xiei(t). The matrices of the operators Ix

y andIyx are equal to the identity matrix in these coordinates.

Let Yi(t) = Y ji (t)ej(t) (1 ≤ i ≤ n) be the Jacobi vector fields along γ satisfying the

initial conditions Yi(0) = 0 and Y ′i (0) = ei(0). Then

(Y j

i (1))

is the matrix of the operatorexp′x(v) in the chosen coordinates because Jacobi vector fields along γ correspond togeodesic variations of γ. Similarly, let Zi(t) = Zj

i (t)ej(t) (1 ≤ i ≤ n) be the Jacobivector fields along γ satisfying the initial conditions Zi(1) = 0 and Z ′

i(1) = ei(1). Then− (

Zji (0)

)is the matrix of exp′y(w).

By the Jacobi equationY ′′j

i + Rj1k1Y

ki = 0,

the Wronskian

W (t) = det

(Y j

i (t) Y ′ji (t)

Zji (t) Z ′j

i (t)

)

is constant, i.e., W (0) = W (1). We can now write the chain of equalities

|det exp′y(w)| = |det(Zj

i (0)) | = |W (0)| = |W (1)| = |det

(Y j

i (1)) | = |det exp′x(v)|

which proves the lemma.Proof of Theorem 6.1. Let u and w be two half-densities supported in the domain

of a local coordinate system, so we can represent them in the form

u = λ(x)|dx|1/2, w = µ(x)|dx|1/2.

By Lemma 6.2, we can rewrite integral (6.1) as follows:

Aw(y) = (2π)−n|dy|1/2

∫

T ∗y X

∫

TyX

e−i〈v,η〉a(y, η)µ(expyv)|det Iyexpyv|1/2dvdη.

Therefore

(Aw, u)L2 =

∫

X

Aw(y)λ(y) |dy|1/2

= (2π)−n

∫

X

∫

T ∗y X

∫

TyX

e−i〈v,η〉a(y, η)µ(expyv)|det Iyexpyv|1/2λ(y)dvdηdy.

36

We change the integration variable v to x = expyv and obtain

(Aw, u)L2 =

∫

X

µ(x)|dx|1/2×

× (2π)−n|dx|1/2

∫

T ∗y X

∫

X

e−i〈exp−1y x,η〉a(y, η)λ(y)|det Iy

x |1/2

∣∣∣∣det∂ exp−1

y x

∂x

∣∣∣∣ dηdy.

From this equality, we see that

A∗u(x) = (2π)−n|dx|1/2

∫

T ∗y X

∫

X

ei〈exp−1y x,η〉a(y, η)λ(y)|det Iy

x |1/2

∣∣∣∣det∂ exp−1

y x

∂x

∣∣∣∣ dydη.

In the latter integral, we change the integration variables (y, η) to (v, ξ) by the formulas

y = expxv, η = tIexpxvx ξ.

Then 〈exp−1y x, η〉 = −〈v, ξ〉, and the Jacobian of the change is

∣∣∣∣det∂(y, η)

∂(v, ξ)

∣∣∣∣ = |det exp′x(v)| · |det Ixexpxv|−1.

After the change, the integral takes the form

A∗u(x) = (2π)−n|dx|1/2

∫

T ∗x X

∫

TxX

e−i〈v,ξ〉a(expxv, tIexpxvx ξ)λ(expxv)χ(x, v)dvdξ,

where

χ(x, v) = |det Ixexpxv|−3/2|det exp′x(v)| ·

∣∣∣∣∣det∂ exp−1

y x

∂x

∣∣∣∣y=expxv

∣∣∣∣∣ .

By Lemma 6.2,

|dx|1/2λ(expxv) = |det Ixexpxv|−1/2Jexpxv

x u(expxv),

and the latter integral can be rewritten as follows:

A∗u(x) = (2π)−n

∫

T ∗x X

∫

TxX

e−i〈v,ξ〉a(expxv, tIexpxvx ξ)(Jexpxv

x u(expxv))κ(x, v)dvdξ, (6.2)

where

κ(x, v) = |det Iexpxvx |2|det exp′x(v)| ·

∣∣∣∣∣det∂ exp−1

y x

∂x

∣∣∣∣y=expxv

∣∣∣∣∣ .

If, for fixed x ∈ X and v ∈ TxX, we denote y = expxv and introduce the vectorw ∈ TyX such that x = expyw, then

∣∣∣∣∣det∂ exp−1

y x

∂x

∣∣∣∣y=expxv

∣∣∣∣∣

−1

= |det exp′y(w)|,

37

and Lemma 6.3 shows us that κ(x, v) = 1.We can now rewrite (6.2) in the form

A∗u(x) = (2π)−n

∫

T ∗x X

∫

TxX

e−i〈v,ξ〉a(expxv, tIexpxvx ξ)Jexpxv

x u(expxv) dvdξ. (6.3)

Applying Lemma 3.1 (more exactly, an analog of the lemma for half-densities), we obtainthe following asymptotic series for the geometric symbol b(x, ξ) of the operator A∗:

b(x, ξ) ∼∑

j

1

j!〈∂v, Dξ〉j

(a(expxv, tIexpxv

x ξ))∣∣∣

v=0=

∑α

1

α!∂α

ξ

(Dα

v a(expxv, tIexpxvx ξ)|v=0

).

Finally, applying (2.20), we obtain the statement of the theorem.

7 Differential operators on vector bundles

Given a vector bundle E over a manifold X and an open set U ⊂ X, by C∞(E; U)we denote the C∞(U)-module of smooth sections of E. The notation C∞(E; X) willbe usually abbreviated to C∞(E). All vector bundles are assumed to be complex andsmooth except for the tangent bundle TX and cotangent bundle T ∗X which are realvector bundles. The fiber of a bundle E over a point x ∈ X is denoted by Ex. Giventwo vector bundles E and F over X, by Hom(E, F ) we denote the vector bundle whosefiber over a point x ∈ X is the space Hom(Ex, Fx) of linear operators from Ex to Fx.In particular, End(E) = Hom(E, E). By E∗ we denote the dual bundle of E, and forϕ ∈ C∞(Hom (E,F )) we denote the dual morphism by tϕ ∈ C∞(Hom (F ∗, E∗)).

Let us remind that τ rs X denotes the bundle of (r, s)-tensors over X. Given a vector

bundle E over X, the bundle E⊗τ rs X is called the bundle of E-valued tensors of the rank

(r, s). Two types of coordinate representation are possible for E-valued tensor fields. Thefirst, most orthodox, coordinate representation is as follows.

Let (x1, . . . , xn) be a local coordinate system with the domain U ⊂ X and (e1, . . . , eN)be a trivialization of the bundle E over U . Then, for an E-valued tensor field u ∈C∞(E⊗ τ r

s X), the coordinates uai1...irj1...js

∈ C∞(U) are defined which are transformed undera change of the local coordinate system and trivialization as follows. If (x′1, . . . , x′n)is a second coordinate system with the domain U ′ ⊂ X and (e′1, . . . , e

′N) is a second

trivialization of E over U ′, then

u′ai1...irj1...js

= εab

∂x′i1

∂xk1. . .

∂x′ir

∂xkr

∂xl1

∂x′j1. . .

∂xls

∂x′jsubk1...kr

l1...ls(7.1)

in U∩U ′; where (εba) is the transformation matrix from the basis (e1, . . . , eN) to (e′1, . . . , e

′N),

i.e., e′a = εbaeb. We use the agreement: the indices a, b, c . . . take values from 1 to N while

the indices i, j, k, . . . , from 1 to n; the corresponding summation is assumed on repeatingindices.

We will mostly use the second, more brief, coordinate representation that is definedas follows. Formula (7.1) implies that the sections ui1...ir

j1...js= uai1...ir

j1...jsea ∈ C∞(E; U) are

38

independent of the choice of a trivialization (but still depend on a local coordinate systemon X). We will consider these sections as coordinates of the field u with respect to alocal coordinate system and will write u = (ui1...ir

j1...js). Under a coordinate change, these

sections are transformed by the same formula (2.1) as coordinates of an ordinary tensorfield. On using this coordinate representation, we can operate analytically with E-valuedtensor fields almost in the same way as with ordinary tensor fields. The only thing weshould remember is that coordinates of a field are not scalar functions anymore but theyare sections of E.

From now on in this section, we assume (X,∇) to be a manifold with a fixed symmetricconnection.

Let E be a vector bundle over X and

∇E : C∞(τX)× C∞(E) → C∞(E), ∇E : (v, u) 7→ ∇Ev u

be a connection on the bundle E. If (x1, . . . , xn) is a local coordinate system with thedomain U ⊂ X and (e1, . . . , eN) is a trivialization of the bundle E over U , then we denotethe coefficients of the connection ∇E by Eb

ai ∈ C∞(U), i.e.,

∇E∂i

ea = Ebaieb, where ∂i =

∂

∂xi.

The connection coefficients are transformed under a change of local coordinates and triv-ialization by the formula

E ′bai =

∂xj

∂x′i(ε−1)b

cEcdjε

da + (ε−1)b

c

∂εca

∂x′i,

where the same notations are used as in (7.1).The connections ∇ and ∇E determine the covariant derivative

∇ : C∞(E ⊗ τ rs X) → C∞(E ⊗ τ r

s+1X)

which is expressed in local coordinates by the formula

∇kuai1...irj1...js

=∂

∂xkuai1...ir

j1...js+

r∑κ=1

Γiκkpu

ai1...iκ−1piκ+1...irj1...js

−s∑

κ=1

Γpkjκ

uai1...irj1...jκ−1pjκ+1...js

+ Eabku

bi1...irj1...js

.

We will mostly write this formula in the matrix form

∇kui1...irj1...js

=∂

∂xkui1...ir

j1...js+

r∑κ=1

Γiκkpu

ai1...iκ−1piκ+1...irj1...js

−s∑

κ=1

Γpkjκ

ui1...irj1...jκ−1pjκ+1...js

+ Ekui1...irj1...js

, (7.2)

39

where ui1...irj1...js

is the N -column (uai1...irj1...js

)Na=1 and Ek is the N ×N -matrix (Ea

bk)Na,b=1. While

using the latter formula, we have to remember that ∂∂xk ui1...ir

j1...jsand Ek make sense only

with respect to a trivialization.By E = (Ea

bij) we denote the curvature tensor of the connection ∇E

Eabij =

∂

∂xiEa

bj −∂

∂xjEa

bi + EaciE

cbj − Ea

cjEcbi.

Given a local coordinate system with the domain U ⊂ X, by Eij ∈ C∞(End E; U) wedenote the curvature operator whose matrix with respect to a trivialization is (Ea

bij)Na,b=1.

This definition is independent of the choice of a trivialization because (Eabij) is a tensor.

Let us remind the commutator formula for second order derivatives: for u = (ui1...irj1...js

) ∈C∞(E ⊗ τ r

s X),

[∇k,∇l]ui1...irj1...js

=r∑

κ=1

Riκpklu

i1...iκ−1piκ+1...irj1...js

−s∑

κ=1

Rpjκklu

i1...irj1...jκ−1pjκ+1...js

+ Eklui1...irj1...js

. (7.3)

Unlike (7.2), all terms on the latter formula are independent of the choice of a trivializa-tion.

Given a multi-index α = 〈j1 . . . jm〉, we let

∇α = σ(j1 . . . jm) (∇j1 . . .∇jm)

denote the symmetrized covariant derivative.The connection ∇ determines the exponential map expx : TxX ⊃ U → X in some

neighborhood of the origin 0 ∈ TxX for any point x ∈ X. Beside this, for a smooth curveγ : [0, 1] → X, the connection ∇E determines the parallel transport Jγ : Eγ(0) → Eγ(1).Given two sufficiently close points x, y ∈ X, we let Jx

y : Ex → Ey denote the paralleltransport along the unique shortest geodesic from x to y. The parallel transport is relatedto the covariant derivative by the following

Lemma 7.1 Let (E,∇E) be a vector bundle with a connection over X. For any sectionu ∈ C∞(E) and for any parameterized curve γ : (a, b) → X,

d

dt

(J

γ(t)γ(0)u(γ(t))

)= J

γ(t)γ(0)

(γj(t)∇ju(γ(t))

),

where Jγ(t)γ(0) : Eγ(t) → Eγ(0) is the parallel transport along γ.

Proof. Choose a basis (e1(t), . . . , eN(t)) of Eγ(t) such that every ea(t) is parallel alongγ. Expand u(γ(t)) in the basis: u(γ(t)) = ϕa(t)ea(t). Then

γj(t)∇ju(γ(t)) =dϕa(t)

dtea(t)

and

Jγ(t)γ(0)

(γj(t)∇ju(γ(t))

)=

dϕa(t)

dtJ

γ(t)γ(0)ea(t) =

dϕa(t)

dtea(0),

Jγ(t)γ(0)u(γ(t)) = ϕa(t)ea(0).

Two last equalities imply the statement of the lemma.

40

Lemma 7.2 Under hypotheses of Lemma 7.1, the equality

∇αu(x) = ∂αv [Jexpxv

x u(expxv)]v=0

holds for any section u ∈ C∞(E) and for any multi-index α.

The proof is omitted because we will prove a more general statement later, see Lemma7.5 below.

Given two vector bundles E and F over a manifold X, let

P (T ∗X, Hom(E, F )) =∞⊕

m=0

Pmhom(T ∗X, Hom(E, F ))

be the graded vector space of smooth functions a : T ∗X → Hom(E, F ), a = a(x, ξ),depending polynomially on ξ and such that a(x, ξ) ∈ Hom(Ex, Fx) for (x, ξ) ∈ T ∗X.Here Pm

hom(T ∗X, Hom(E, F )) is the space of homogeneous polynomials in ξ of degree m.The space of polynomials of degree ≤ m is denoted by Pm(T ∗X, Hom(E, F )). Such apolynomial a ∈ Pm(T ∗X, Hom(E, F )) can be written in the form

a(x, ξ) =∑

|α|≤m

aα(x)ξα

in local coordinates, where aα(x) ∈ Hom(Ex, Fx). Let

DO(E, F ) =∞⋃

m=0

DOm(E, F )

be the space of differential operators A : C∞(E) → C∞(F ) from E to F . Here DOm(E, F )is the space of differential operators of order ≤ m.

Let now (E,∇E) be a vector bundle with a connection and F be a second vectorbundle over X. The isomorphism

P (T ∗X, Hom(E, F )) −→ DO(E,F ), a(x, ξ) 7→ a(x,−i∇),

being defined in coordinates by the formula

a(x,−i∇) =∑

|α|≤m

aα(x)(−i∇)α for a(x, ξ) =∑

|α|≤m

aα(x)ξα,

is independent of the choice of a local coordinate system. We will say that the polynomiala(x, ξ) is the geometric symbol of the differential operator a(x,−i∇) with respect to theconnections ∇ and ∇E. Lemma 7.2 implies immediately

Corollary 7.3 For a polynomial a ∈ P (T ∗X,Hom(E,F )) and section u ∈ C∞(E),

a(x,−i∇)u(x) = a(x,Dv) [Jexpxvx u(expxv)]v=0 ,

where Dv = −i∂v.

41

A connection on E determines uniquely the connection on the dual bundle E∗ by therequirement that the covariant derivative is agreed with the pairing of E and E∗. If(E,∇E) and (F,∇F ) are two bundles with connections, then E ⊗F is furnished with theconnection that is agreed with the tensor product. Therefore the bundle Hom(E, F ) ∼=E∗ ⊗ F is canonically furnished with the connection.

Let now (E,∇E), (F,∇F ) be two bundles with connections and G be a third bundleover X. Given a polynomial a ∈ P (T ∗X, Hom(F, G)), the differential operator

a(x,−i∇) : C∞(F ) → C∞(G)

is also determined on sections of the bundle Hom(E, F )

a(x,−i∇) : C∞(Hom(E,F )) → C∞(Hom(E, G)).

Indeed, in local coordinates

a(x,−i∇) =∑

|α|≤m

aα(x)(−i∇)α, aα ∈ C∞(Hom(F,G)).

As was mentioned in the previous paragraph, ∇αϕ makes sense for ϕ ∈ C∞(Hom(E, F )).Therefore

a(x,−i∇)ϕ =∑

|α|≤m

aα · (−i∇)αϕ.

The summand of the latter sum is the product of ∇αϕ ∈ C∞(Hom(E,F )) and aα ∈C∞(Hom(F,G)). Therefore aα · ∇αϕ ∈ C∞(Hom(E, G)).

Lemma 7.4 (Leibnitz formula) Let (E,∇E), (F,∇F ) be two bundles with connections,and G be a third bundle over X. For a ∈ P (T ∗X,Hom(F, G)), ϕ ∈ C∞(Hom(E, F )) andu ∈ C∞(E), the equality

a(x,−i∇)(ϕu) =∑

α

1

α!(

v

∇αa)(x,−i∇)ϕ · (−i∇)αu

holds. Hereafterv

∇α = ∂αξ .

Proof. It suffices to consider the case of a(x, ξ) = ξλ. In this casev

∇αa = λ!(λ−α)!

ξλ−α,and the equality we are going to prove looks as follows:

(−i∇)λ(ϕu) =∑

α

(λα

)(−i∇)λ−αϕ · (−i∇)αu.

By Lemma 7.2,

(−i∇)λ(ϕu)(x) = Dλv [Jexpxv

x (ϕu)(expxv)]v=0 = Dλv [Jexpxv

x (ϕ(expxv) · u(expxv))]v=0 .

The parallel transport is agreed with the product, i.e.,

Jexpxvx (ϕ(expxv) · u(expxv)) = Jexpxv

x (ϕ(expxv)) · Jexpxvx (u(expxv)) .

42

Therefore

(−i∇)λ(ϕu)(x) = Dλv [Jexpxv

x (ϕ(expxv)) · Jexpxvx (u(expxv))]v=0 .

Applying the classical Leibnitz formula, we have

(−i∇)λ(ϕu)(x) =∑

α

(λα

)Dλ−α

v [Jexpxvx (ϕ(expxv))]v=0 ·Dα

v [Jexpxvx (u(expxv))]v=0 .

Using Lemma 7.2 again, we get the desired statement.

We are going to derive the formula for the geometric symbol of the product of twodifferential operators. To this end, given a bundle (E,∇E) with a connection, we introducepolynomials Rα,β(x, ξ) ∈ P (T ∗X, End(E)) by the equalities

(−i∇)α(−i∇)β = Rα,β(x,−i∇). (7.4)

These polynomials play the role of the multiplication table in the algebra DO(E,E) withrespect to the basis (−i∇)α. The leading term of the polynomial Rα,β(x, ξ) is ξα+β. Moreprecisely, Rα,β(x, ξ) = ξα+β + . . . , where dots mean a polynomial of degree ≤ |α|+ |β|−2.Several first polynomials Rα,β are as follows:

Rα,0 = R0,α = ξα, (7.5)

R〈j〉,〈k〉 = ξjξk − 1

2Ejk, (7.6)

R〈j〉,〈kl〉 = ξjξkξl +1

3(Rp

kjl + Rpljk)ξp − 1

2(Ejkξl + Ejlξk)− 1

6((−i∇)kEjl + (−i∇)lEjk) ,

(7.7)

R〈jk〉,〈l〉 = ξjξkξl +1

6(Rp

jkl + Rpkjl)ξp − 1

2(Ejlξk + Eklξj)− 1

3((−i∇)jEkl + (−i∇)kEjl) ,

(7.8)

R〈j〉,〈klm〉 = ξjξkξlξm +

1

4σ(klm)

(8Rp

kjlξmξp − 6Ejkξlξm

+ 2(−i∇)kRpljmξp − 4(−i∇)kEjlξm − (−i∇)k(−i∇)lEjm + Rp

kljEmp

), (7.9)

R〈jkl〉,〈m〉 = ξjξkξlξm +1

4σ(jkl)

(4Rp

jkmξlξp − 6Ejmξkξl

+ 2(−i∇)iRpklmξp − 8(−i∇)jEkmξl − 3(−i∇)j(−i∇)kElm −Rp

jkmElp

), (7.10)

R〈jk〉,〈lm〉 = ξjξkξlξm +1

6σ(jk)σ(lm)

(8Rp

ljmξkξp + 4Rpjklξmξp − 12Ejlξkξm

+ 5(−i∇)jRplkmξp + (−i∇)lR

pjkmξp − 8(−i∇)jEklξm − 4(−i∇)lEjmξk

−3(−i∇)j(−i∇)lEkm + 2RplmjEkp + Rp

jklEpm + 3EjlEkm

), (7.11)

In the particular case of E = 0, these formulas coincide with (2.12)–(2.17). In principle,any polynomial Rα,β can be written down explicitly in terms of the curvature tensors Rand E by repeatedly using the commutator formula (7.2).

43

Recall that π : T ∗X → X is the cotangent bundle. The pull-back βrsX = π∗(τ r

s X) isa vector bundle over T ∗X which is called the bundle of semibasic tensors of rank (r, s).Recall that the connection ∇ allows us to define the horizontal covariant derivative

h

∇ : C∞(βrsX) → C∞(βr

s+1X) (7.12)

The definitions of the previous paragraph can be generalized as follows. Given avector bundle E over X, we define the bundle βr

s(X,E) = π∗(E ⊗ τ rs X) over T ∗X which

is called the bundle of E-valued semibasic tensors. Sections of the bundle are called E-valued semibasic tensor fields. In particular, a section u ∈ C∞(β0

0(X, E)) is a functionu : T ∗X → E such that u(x, ξ) ∈ Ex for (x, ξ) ∈ T ∗X. For a field u ∈ C∞(βr

s(X, E)),the coordinates uai1...ir

j1...js(x, ξ) are defined in the domain of a local coordinate system and

trivialization which are transformed according to the same formula (7.1) as coordinates ofan ordinary E-valued tensor field. From analytical viewpoint, the only difference betweenordinary E-valued tensor fields and semibasic E-valued tensor fields is that the coordinatesof a latter field depend not only on a point x ∈ X but also on a covector ξ ∈ T ∗

xX. Aconnection∇E on E allows us to generalize the horizontal derivative (7.12) to the operator

h

∇ : C∞(βrs(X,E)) → C∞(βr

s+1(X, E)).

In local coordinates the operator is defined by the formula

h

∇kuai1...irj1...js

=∂

∂xkuai1...ir

j1...js+ Γq

kpξq∂

∂ξp

uai1...irj1...js

+r∑

κ=1

Γiκkpu

ai1...iκ−1piκ+1...irj1...js

−s∑

κ=1

Γpkjκ

uai1...irj1...jκ−1pjκ+1...js

+ Eabku

bi1...irj1...js

.

h

∇ commutes with the vertical derivativev

∇ = ∂ξ : C∞(βrs(X,E)) → C∞(βr+1

s (X,E)).The commutator formula for horizontal derivatives looks as follows:

[h

∇k,h

∇l]ui1...irj1...js

= Rqpklξq

v

∇pui1...irj1...js

+r∑

κ=1

Riκpklu

i1...iκ−1piκ+1...irj1...js

−s∑

κ=1

Rpjκklu

i1...irj1...jκ−1pjκ+1...js

+ Eklui1...irj1...js

. (7.13)

For the horizontal derivative, we will use the same notation as above:

h

∇α = σ(j1 . . . jm)

(h

∇j1 . . .h

∇jm

)for a multi-index α = 〈j1 . . . jm〉.

Lemma 7.5 Let (E,∇E) be a vector bundle with connection over X. The equality

h

∇αu(x, ξ) = ∂αv

(Jexpxv

x u(expxv, t(Ixexpxv)

−1ξ))∣∣∣

v=0(7.14)

holds for any u ∈ C∞(β00(X, E)) and for any multi-index α.

44

In particular, this gives Lemma 7.2 in the case of u = u(x) independent of ξ.Proof. Fix a point (x0, ξ0) ∈ T ∗X and vector w0 ∈ Tx0X. Let γ(t) = expx0

tw0.Choose a section w ∈ C∞(TX; U) of the tangent bundle over a sufficiently small neigh-borhood U of the point x0 such that w(γ(t)) = γ(t) for small t. Define the sectionη ∈ C∞(T ∗X; U) of the cotangent bundle over U by putting η(x) = t(Ix0

x )−1ξ0. In partic-ular, w(x0) = w0 and η(x0) = ξ0. For every m = 0, 1, . . . define the section em ∈ C∞(E; U)of the bundle E over U by putting

em(x) = (wj1 . . . wjmh

∇j1 . . .h

∇jmu)(x, η(x)). (7.15)

These sections satisfy the relation

em+1(γ(t)) = γj(t)∇jem(γ(t)). (7.16)

Assume for a minute (7.16) to be proved. Applying the operator Jγ(t)x0 to this equation,

we obtain

Jγ(t)x0

em+1(γ(t)) = Jγ(t)γ(0)(γ

j(t)∇jem(γ(t))) =d

dt

(Jγ(t)

x0em(γ(t))

),

where the last equality is written by Lemma 7.1. We have thus proved that

d

dt

(Jγ(t)

x0em(γ(t))

)= Jγ(t)

x0em+1(γ(t)),

and thereforedm

dtm

(Jγ(t)

x0e0(γ(t))

)= Jγ(t)

x0em(γ(t)).

Substitute the values γ(t) = expx0tw0 and e0(γ(t)) = u(expx0

tw0,t(Ix0

expx0tw0

)−1ξ0) into the

latter formula

dm

dtm

(J

expx0tw0

x0 u(expx0tw0,

t(Ix0expx0

tw0)−1ξ0)

)= Jγ(t)

x0em(γ(t)).

Set t = 0

dm

dtm

(J

expx0tw0

x0 u(expx0tw0,

t(Ix0expx0

tw0)−1ξ0)

)∣∣∣∣t=0

= em(x0) = wj10 . . . wjm

0

h

∇j1 . . .h

∇jmu(x0, ξ0).

On the other hand,

dm

dtm

(J

expx0tw0

x0 u(expx0tw0,

t(Ix0expx0

tw0)−1ξ0)

)∣∣∣∣t=0

= wj10 . . . wjm

0 ∂vj1 . . . ∂vjm

(J

expx0v

x0 u(expx0v, t(Ix0

expx0v)−1ξ0)

)∣∣∣v=0

.

Comparing the two last formulas, we obtain

wj10 . . . wjm

0

h

∇j1 . . .h

∇jmu(x0, ξ0)

= wj10 . . . wjm

0 ∂vj1 . . . ∂vjm

(J

expx0v

x0 u(expx0v, t(Ix0

expx0v)−1ξ0)

)∣∣∣v=0

.

45

This equation is equivalent to (7.14) because w0 ∈ Tx0X is an arbitrary vector.It remains to prove (7.16). To this end we apply the operator ∇w = wk∇k = wk ∂

∂xk +wkEk to equation (7.15)

(wk∇kem)(x) = wj1 . . . wjm

(wk ∂

∂xk

(h

∇j1 . . .h

∇jmu(x, η(x))

)+ wkEk

h

∇j1 . . .h

∇jmu(x, η(x))

)

+m∑

λ=1

wj1 . . . wjλ−1wk ∂wjλ

∂xkwjλ+1 . . . wjm

h

∇j1 . . .h

∇jmu(x, η(x)).

Substituting the value

∂

∂xk

(h

∇j1 . . .h

∇jmu(x, η(x))

)=

(∂

∂xk

h

∇j1 . . .h

∇jmu

)(x, η(x))

+∂ηp(x)

∂xk

(∂

∂ξp

h

∇j1 . . .h

∇jmu

)(x, η(x))

into the latter equation, we have

wk∇kem = wj1 . . . wjm

(wk ∂

∂xk

h

∇j1 . . .h

∇jmu + wk ∂ηp

∂xk

∂

∂ξp

h

∇j1 . . .h

∇jmu + wkEk

h

∇j1 . . .h

∇jmu

)

+m∑

λ=1

wj1 . . . wjλ−1wk ∂wjλ

∂xkwjλ+1 . . . wjm

h

∇j1 . . .h

∇jmu. (7.17)

For brevity, arguments are omitted here but all terms are assumed to be taken at thepoint (x, η(x)). In particular, for x = γ(t),

wk ∂ηp

∂xk= γk ∂ηp

∂xk=

dηp(γ(t))

dt= wkΓq

kpηq (7.18)

because the covector η(γ(t)) is parallel along γ(t). Beside this, for x = γ(t),

wk ∂wjλ

∂xk= γk ∂wjλ

∂xk= γjλ(t) = −Γjλ

kpwkwp. (7.19)

Substitute the values (7.18) and (7.19) into (7.17) to obtain

(wk∇kem)(γ(t)) =

[wkwj1 . . . wjm

( ∂

∂xk

h

∇j1 . . .h

∇jmu + Γqkpηq

∂

∂ξp

h

∇j1 . . .h

∇jmu

+ Ek

h

∇j1 . . .h

∇jmu)−

m∑

λ=1

wj1 . . . wjλ−1Γjλ

kpwkwpwjλ+1 . . . wjm

h

∇j1 . . .h

∇jmu

]

x=γ(t)

ξ=η(γ(t))

.

After transposing the summation indices jλ and p in the last sum, this equation takes theform

(wk∇kem)(γ(t)) =

[wkwj1 . . . wjm

( ∂

∂xk

h

∇j1 . . .h

∇jmu + Γqkpξq

∂

∂ξp

h

∇j1 . . .h

∇jmu +

46

+ Ek

h

∇j1 . . .h

∇jmu−m∑

λ=1

Γpkjλ

h

∇j1 . . .h

∇jλ−1

h

∇p

h

∇jλ+1. . .

h

∇jmu)]

x=γ(t)

ξ=η(γ(t))

.

We recognize the horizontal derivativeh

∇k

h

∇j1 . . .h

∇jmu in the parentheses and write thelatter equation in the form

(wk∇kem)(γ(t)) =

(wj1 . . . wjm+1

h

∇j1 . . .h

∇jm+1u

)(γ(t), η(t)).

This is equivalent to (7.16) because w(γ(t)) = γ(t). The lemma is proved.

Formula (7.14) has the following corollary: for b ∈ C∞(β00(X, E)), the representation

∂αv

(Jexpxv

x

v

∇βb(expxv, t(Ixexpxv)

−1ξ))∣∣∣

v=0=

∑γ≤α

∑

|δ|=|β|

h

∇γv

∇δb(x, ξ) · pαβγδ (x) (7.20)

holds for any multi-indices α and β, where pαβγδ (x) are some smooth functions in the

domain of a local coordinate system which are independent of b and ξ. Indeed, fix local

coordinates in a neighborhood of x and put u(x, ξ) =v

∇βb(x, ξ) for a given multi-index β.With the help of induction on |α|, one easily proves the representation

h

∇αu(x, ξ)− h

∇αv

∇βb(x, ξ) =∑γ<α

∑

|δ|=|β|

h

∇γv

∇δb(x, ξ) · pαβγδ (x)

which, together with (7.14), gives (7.20).

Let now (E,∇E) and (F,∇F ) be two bundles with connections over X. A polynomiala(x, ξ) =

∑α

aα(x)ξα ∈ P (T ∗X, Hom(E, F )) can be considered as a Hom(E, F )-valued

semibasic tensor field of rank (0, 0). The following equality holds:

h

∇β

(∑α

aα(x)ξα

)=

∑α

(∇βaα(x))ξα. (7.21)

Theorem 7.6 Let (X,∇) be a manifold with connection; (E,∇E), (F,∇F ) be two vectorbundles with connections; and G be a third vector bundle over X. Given polynomialsa ∈ P (T ∗X,Hom(F, G)) and b ∈ P (T ∗X,Hom(E, F )), the geometric symbol of the product

c(x,−i∇) = a(x,−i∇)b(x,−i∇)

of two differential operators is expressed through a(x, ξ) and b(x, ξ) by the formula

c(x, ξ) =∑

α

1

α!

v

∇αa∑

β,γ

1

γ!

(αβ

)(−i

h

∇)βv

∇γb · ρα−β,γ, (7.22)

47

where

(αβ

)= α!

β!(α−β)!are the binomial coefficients with

(αβ

)6= 0 only for β ≤ α; and

ρα,β ∈ P (T ∗X,End(E)) are polynomials expressed through polynomials (7.4) by the for-mula

ρα,β = (−1)|α|+|β|∑

λ,µ

(−1)|λ|+|µ|(

αλ

) (βµ

)ξα+β−λ−µRλ,µ. (7.23)

The degree of the polynomial ρα,β is ≤ |α|/4 + 3|β|/4.

Before proving the theorem, we will make some remarks.The coefficients ρα,β depend on ∇ and ∇E but not on ∇F . The connection ∇F

participates in (7.22) through the horizontal derivativesh

∇βb.ρ0,0 = 1 and ρα,β = 0 for |α|+|β| > 0 in the case of flat connections when the curvature

tensors are equal to zero, and formula (2.19) coincides with the classical one:

c(x, ξ) =∑

α

1

α!∂α

ξ a ·Dαx b.

As follows from (7.5)–(7.11) and (7.23), several first polynomials ρα,β are

ρ0,0 = 1, ρα,0 = ρ0,α = 0 for |α| > 0, (7.24)

ρ〈j〉,〈k〉 = −1

2Ejk, (7.25)

ρ〈j〉,〈kl〉 = −1

3(Rp

klj + Rplkj)ξp − 1

6((−i∇)kEjl + (−i∇)lEjk) , (7.26)

ρ〈jk〉,〈l〉 = −1

6(Rp

jlk + Rpklj)ξp − 1

3((−i∇)jEkl + (−i∇)kEjl) , (7.27)

ρ〈j〉,〈klm〉 =1

4σ(klm)

(2(−i∇)kR

pljmξp − (−i∇)k(−i∇)lEjm + Rp

kljEmp

), (7.28)

ρ〈jkl〉,〈m〉 =1

4σ(jkl)

(2(−i∇)jR

pklmξp − 3(−i∇)j(−i∇)kElm −Rp

jkmElp

), (7.29)

ρ〈jk〉,〈lm〉 =1

6σ(jk)σ(lm)

(5(−i∇)jR

plkmξp + (−i∇)lR

pjkmξp

−3(−i∇)j(−i∇)lEkm + 2RplmjEkp + Rp

jklEpm + 3EjlEkm

). (7.30)

The proof of Theorem 7.6 does not coincide with the proof of Theorem 2.4 becausethe factors ϕ and u on the left-hand side of the Leibnitz formula (Lemma 7.4) are notequal in rights unlike the scalar case in Corollary 2.3.

Proof of Theorem 7.6. Let

a(x, ξ) =∑

λ

aλ(x)ξλ, b(x, ξ) =∑

µ

bµ(x)ξµ.

48

For u ∈ C∞(E),

c(x,−i∇)u(x) = a(x,−i∇)(b(x,−i∇)u(x)) = a(x,−i∇)

(∑µ

bµ(x)(−i∇)µu(x)

).

Using the Leibnitz formula and (7.4), we obtain

c(x,−i∇)u(x) =∑α,µ

1

α!(

v

∇αa)(x,−i∇)bµ(x) · (−i∇)α(−i∇)µu(x)

=∑α,µ

1

α!(

v

∇αa)(x,−i∇)bµ(x) ·Rα,µ(x,−i∇)u(x).

We have thus proved that

c(x,−i∇) =∑α,µ

1

α!(

v

∇αa)(x,−i∇)bµ(x) ·Rα,µ(x,−i∇).

Therefore

c(x, ξ) =∑α,µ

1

α!(

v

∇αa)(x,−i∇)bµ(x) ·Rα,µ(x, ξ).

Substituting the expression

v

∇αa(x,−i∇) =∑

λ

λ!

(λ− α)!aλ(x)(−i∇)λ−α

into the previous formula, we obtain

c(x, ξ) =∑

λ

(λα

)aλ(x)(−i∇)λ−αbµ(x) ·Rα,β(x, ξ). (7.31)

Let us prove equivalence of the formulas (7.22) and (7.31). To this end, we substitutethe expressions

v

∇αa =∑

λ

λ!

(λ− α)!aλ(x)ξλ−α, (−i

h

∇)βv

∇γb =∑

µ

µ!

(µ− γ)!(−i∇)βbµ(x)ξµ−γ

into (7.22) to obtain

c(x, ξ) =∑

α,β,γ,λ,µ

(λα

)(µγ

)(αβ

)aλ(x)(−i∇)βbµ(x) · ρα−β,γ(x, ξ)ξλ−αξµ−γ.

We write the latter equality in the form

c(x, ξ) =∑

β,λ,µ

aλ(x)(−i∇)βbµ(x) ·∑α,γ

(λα

)(µγ

)(αβ

)ρα−β,γξλ−αξµ−γ.

49

After the change β = λ− α, formula (7.31) takes the form

c(x, ξ) =∑

β,λ,µ

aλ(x)(−i∇)βbµ(x) ·(

λβ

)Rλ−β,β(x, ξ).

Comparing two last formulas, we see that the proof is reduced to checking the equality

∑α,γ

(λα

)(µγ

)(αβ

)ρα−β,γξλ−αξµ−γ =

(λβ

)Rλ−β,β.

The latter equality follows from Lemma 2.5. Moreover, this equality determines uniquelythe polynomials ρα,β. The theorem is proved.

We finish the section with discussing differential operator on vector-valued half-densities.Let us remind that Ω1/2X is the bundle of half-densities over X. Given a vector bundleE over X, E ⊗ Ω1/2X is the bundle of E-valued half-densities. Locally, in the domain Uof a coordinate system, an E-valued half-density u ∈ C∞(E ⊗Ω1/2X) can be representedas u = λ(x)|dx|1/2 with a section λ ∈ C∞(E; U) which is transformed by formula (2.34)under a coordinate change. If moreover a trivialization (e1, . . . , eN) of E is chosen overU , we can treat λ(x) as an N -column (λa(x))N

a=1, i.e., λ(x) = λa(x)ea(x).Let E and F be two vector bundles over X. A differential operator of order ≤ m

A : C∞(E ⊗ Ω1/2X) → C∞(F ⊗ Ω1/2X)

between half-densities can be analytically written as follows. Let (x1, . . . , xn) be a localcoordinate system with the domain U ⊂ X, (e1, . . . , eN) and (f1, . . . , fM) be trivializationsof the bundles E and F respectively over U . Then

A(λ(x)|dx|1/2) = a(x,D)λ(x) · |dx|1/2 (7.32)

with a polynomial a(x, ξ) =∑|α|≤m

aα(x)ξα whose coefficients are N × M -matrices. Two

leading terms a0(x, ξ) =∑|α|=m

aα(x)ξα and a1(x, ξ) =∑

|α|=m−1

aα(x)ξα of the polynomial

are transformed under a change of coordinates and trivializations by the formulas

a′0(x′, ξ′) = ϕ−1(x)a0(x, ξ)ε(x), (7.33)

a′1(x′, ξ′) = ϕ−1(x)(a1(x, ξ)ε(x)− i

v

∇j a0(x, ξ) · ∂jε(x)

− i

2

∂2x′p

∂xj∂xkξ′p

v

∇jv

∇ka0(x, ξ) · ε(x)− i

2J −1 ∂J

∂xj

v

∇j a0(x, ξ) · ε(x)). (7.34)

Compare with (2.37)–(2.38). Here ε = (εba)

Na,b=1 is the transformation matrix from the

trivialization (e1, . . . , eN) to (e′1, . . . , e′N), i.e., e′a = εb

aeb; and ϕ = (ϕba)

Ma,b=1 is the trans-

formation matrix from (f1, . . . , fM) to (f ′1, . . . , f′M).

50

Let now (E,∇E) be a bundle with connection, and F be a second vector bundle overX. In Section 2, we have defined the covariant derivative (2.42) of tensor half-densities.The latter, together with ∇E, determines the covariant derivative

∇ : C∞(E ⊗ τ 0s X ⊗ Ω1/2X) → C∞(E ⊗ τ 0

s+1X ⊗ Ω1/2X) (7.35)

of E-valued tensor half-densities. Then we define the symmetrized covariant derivative∇αu of an E-valued half-density u ∈ C∞(E ⊗ Ω1/2X) by (2.8). Now, the vector spaceisomorphism

Pm(T ∗X, Hom (E, F )) → DOm(E ⊗ Ω1/2X,F ⊗ Ω1/2X), a(x, ξ) 7→ a(x,−i∇)

is defined by (2.10). As above, the polynomial a(x, ξ) is called the geometric symbolof the operator A = a(x,−i∇) (with respect to the connections ∇ and ∇E), while thepolynomial a(x, ξ) from (7.32) is called the coordinate-wise symbol of A with respect to acoordinate system and trivializations. Two leading terms of these polynomials are relatedby the formulas

a0 = a0, (7.36)

a1 = a1 − i

2Γp

jkξp

v

∇jv

∇ka0 +i

2

v

∇j a0 · (2Ej − Γj). (7.37)

Compare with (2.43)–(2.44). If the bundle F is endowed with a connection ∇F , we candefine the subprincipal symbol

a1s = a1 +i

2

h

∇j

v

∇ja0. (7.38)

Compare with (2.45). With the help of (7.36)–(7.37), one obtains the following expressionfor the subprincipal symbol through the coordinate-wise symbol:

a1s = a1 +i

2

∂2a0

∂xj∂ξj

+i

2(

v

∇j a0 · Ej + Fj ·v

∇j a0). (7.39)

Compare with (2.39). We remind that Ej = (Eabj)

Na,b=1 and Fj = (F a

bj)Ma,b=1 are the

connection matrices of ∇E and ∇F respectively, see (7.2). From (7.39), we see that thesubprincipal symbol is independent of the choice of a connection ∇ on the manifold Xbut depends on the connections ∇E and ∇F .

8 Pseudodifferential operators on vector bundles

Given a vector bundle E over a manifold X and m ∈ R, the space Sm(T ∗X,E) ofsymbols of order ≤ m consists of smooth functions a : T ∗X → E, a = a(x, ξ) such thata(x, ξ) ∈ Ex for (x, ξ) ∈ Ex and the estimate

|∂αx ∂β

ξ a(x, ξ)| ≤ CK,α,β(1 + |ξ|)m−|β| (x ∈ K, ξ ∈ T ∗xX)

holds in any local coordinate system for any multi-indices α, β and for any compact K ⊂ Xwhich is contained in the domain of the coordinate system. The space is invariantly

51

defined, i.e., it is independent of the choice of an atlas on X. Note that Sm(T ∗X, E) ⊂C∞(β0

0(X, E)).Let now (X,∇) be a manifold with a fixed symmetric connection, (E,∇E) be a vector

bundle with a fixed connection, and F be a second vector bundle over X. Given a symbola ∈ Sm(T ∗X, Hom(E, F )), we say that a linear continuous operator

A = a(x,−i∇) : C∞(E) → D′(F )

belongs to Ψm(X,∇; E, F ) and has the geometric symbol a if the Schwartz kernel of A issmooth outside the diagonal and, for every point x ∈ X, there exists a neighborhood Uof x such that

Au(x) = (2π)−n

∫

T ∗x X

∫

TxX

e−i〈v,ξ〉a(x, ξ)Jexpxvx u(expxv)dvdξ (8.1)

for any section u ∈ C∞(E) with supp u ⊂ U . Here Jexpxvx : Eexpxv → Ex is the parallel

transport along the geodesic t 7→ expxv which is determined by the connection ∇E. Notethat the integrand a(x, ξ)J

expxvx u(expxv) belongs to the vector space Fx, so the integral is

well defined.If the symbol a(x, ξ) depends polynomially on ξ, then (8.1) coincides with the differ-

ential operator a(x,−i∇) defined in the previous section. This fact can be easily provedwith the help of Corollary 7.3.

Let Ψm(X; E, F ) be the space of pseudodifferential operators from E to F accordingto the classical definition. The equality

Ψm(X,∇; E, F )/Ψ−∞(X; E, F ) = Ψm(X; E,F )/Ψ−∞(X; E, F )

is proved along with the same line as we have used for proving (3.5) in Sections 3 and 4.In particular, the following generalization of Lemma 3.1 holds:

Lemma 8.1 Let (X,∇) be a manifold with a connection, (E,∇E) be a vector bun-dle with a connection, and F be a second vector bundle over X. For b ∈ Sm(TX ⊕T ∗X,Hom(E, F )), let the operator A be defined by

Au(x) = (2π)−n

∫

T ∗x X

∫

TxX

e−i〈v,ξ〉b(x, v, ξ)Jexpxvx u(expxv)dvdξ

for u ∈ C∞(E) supported in a sufficiently small neighborhood of x. If moreover theSchwartz kernel of A is smooth outside the diagonal, then A ∈ Ψm(X,∇; E, F ) with thegeometric symbol

a(x, ξ) = (2π)−n

∫

T ∗x X

∫

TxX

ei〈v,ξ−η〉b(x, v, η)dvdη

which has the asymptotic expansion

a(x, ξ) ∼∑

j

1

j!〈∂v, Dξ〉jb(x, v, ξ)|v=0.

52

The symbol a(x, ξ) can be recovered from the operator A = a(x,−i∇) by the formula:for e ∈ Ex,

a(x, ξ)e = Ay

(ψ(y)ei〈exp−1

x y,ξ〉Jxy e

)∣∣∣y=x

(mod S−∞(T ∗X, Hom (E, F ))

). (8.2)

Here ψ ∈ C∞0 (X) is a function supported in a sufficiently small neighborhood of the point

x and equal to unity in a smaller neighborhood of x; Ay means the operator A acting withrespect to the variable y, i.e., x and ξ are considered as parameters on the right-hand sideof (8.2). The proof of (8.2) is quite similar to the proof of (3.9) and so omitted.

Theorem 4.1 has the following generalization.

Theorem 8.2 Let a symmetric connection ∇ be fixed on an open set X ⊂ Rn, thetrivial bundle E = X × CN be endowed with a connection ∇E, and F = X × CM . Letb ∈ Sm(X × Rn;Hom(CN ,CM)) and the operator A ∈ Ψm(X; E, F ) is defined by theformula

Au(x) = (2π)−n

∫

Rn

∫

Rn

ei〈x−y,η〉a(x, η)u(y)dydη (8.3)

for u ∈ C∞0 (E). Then A, considered as an operator from Ψm(X,∇; E, F ), has the geo-

metric symbol a(x, ξ) that is expressed through a(x, ξ) by the asymptotic series

a(x, ξ) ∼∑

α,β

1

α!

(αβ

)ψα−β(x, ξ) · v

∇αa(x, ξ) · χβ(x), (8.4)

where the functions ψα(x, ξ) are defined by formulas (4.10)–(4.11) while the matrix-valuedcoefficients χα ∈ C∞(X;Hom(CN ,CN)) are defined by the formula

χα(x) = Dαz Jx

x+z|z=0. (8.5)

The proof is omitted because it is quite similar to the proof of Theorem 4.1. Severalfirst coefficients χα are as follows:

χ0 = I, χ〈j〉 = iEj, χ〈jk〉 = σ(jk)(∂xjEk − EjEk). (8.6)

Equation (8.4) has the following equivalent form

a(x, ξ) ∼∞∑

k=0

1

k!

k∑

l=0

(kl

)ψ〈j1...jl〉(x, ξ) · v

∇j1 . . .v

∇jk a(x, ξ) · χ〈jl+1...jk〉(x) (8.7)

which is sometimes more comfortable for usage.Now, under hypotheses of Theorem 8.2, let us consider a classical pseudodifferential

operator A ∈ Ψmcl (X; E, F ). In this case the symbol a(x, ξ) from (8.3) has the asymptotic

expansiona(x, ξ) ∼ a0(x, ξ) + a1(x, ξ) + a2(x, ξ) + . . . ,

where aj(x, ξ) is a positively homogeneous of degree m− j function in ξ for |ξ| ≥ 1. Thesimilar expansion

a(x, ξ) ∼ a0(x, ξ) + a1(x, ξ) + a2(x, ξ) + . . .

53

holds for the geometric symbol. On using (4.13)–(4.15) and (8.5)–(8.7), one easily obtainsthe formulas

a0(x, ξ) = a0(x, ξ), (8.8)

a1(x, ξ) = a1(x, ξ)− i

2Γp

jk(x)ξp

v

∇jv

∇ka0(x, ξ) + iv

∇j a0(x, ξ) · Ej(x), (8.9)

a2 = a2 − i

2Γp

jkξp

v

∇jv

∇ka1 + iv

∇j a1 · Ej +1

2

v

∇jv

∇ka0(∂xjEk − EjEk) +1

2Γp

jkξp

v

∇jv

∇kv

∇la0 · El

− 1

6(∂xlΓp

jk + ΓpjqΓ

qkl)ξp

v

∇jv

∇kv

∇la0 − 1

8Γp

jkΓqlmξpξq

v

∇jv

∇kv

∇lv

∇ma0, (8.10)

which generalize formulas (1.4), (1.6)–(1.7) to the case of vector bundles.Theorem 5.1 is generalized to the case of vector bundles almost word by word:

Theorem 8.3 Let (X,∇) be a manifold with connection, (E,∇E) and (F,∇F ) be twovector bundles with connections, and G be a third vector bundle over X. Let one of twooperators A = a(x,−i∇) ∈ Ψm1(X,∇; F, G) and B = b(x,−i∇) ∈ Ψm2(X,∇; E, F ) beproperly supported. Then the product C = AB belongs to Ψm1+m2(X,∇; E, G) and thegeometric symbol c(x, ξ) of C is expressed through a(x, ξ) and b(x, ξ) by the asymptoticseries (5.1) whose coefficients ρα,β ∈ P (T ∗X,End(E)) are defined by (7.23).

Sketch of the proof. It goes along with the same line as the proof of Theorem 5.1,so we present only key formulas.

The operator A is given by (8.1), and B is given by the same formula with b substitutedfor a. Inserting one of these formulas into the other, we obtain

(2π)2nABu(x) =

∫

T ∗x

∫

Tx

∫

T ∗expxv

∫

Texpxv

e−i(〈v,ξ〉+〈w,η〉)a(x, ξ)Jexpxvx

(b(expxv, η)J

expexpxvwexpxv u(expexpxvw)

)dwdηdvdξ.

(8.11)Since the parallel transport is agreed with the product,

Jexpxvx

(b(expxv, η)J

expexpxvwexpxv u(expexpxvw)

)

= Jexpxvx b(expxv, η) · Jexpxv

x Jexpexpxvwexpxv u(expexpxvw). (8.12)

Introducing the operator

J(x; v, w) = Jexpxvx J

expexpxvwexpxv Jx

expexpxvw,

we rewrite (8.12) in the form

Jexpxvx

(b(expxv, η)J

expexpxvwexpxv u(expexpxvw)

)

= Jexpxvx b(expxv, η) · J(x; v, w)J

expexpxvwx u(expexpxvw). (8.13)

54

The operator J(x; v, w) : Ex → Ex is the parallel transport along the geodesic trianglewith the vertices x, expxv, and expexpxvw. This operator is determined by the curvature

tensor E of the connection ∇E. Substituting (8.13) into (8.11), we have

(2π)2nABu(x) =∫

T ∗x

∫

Tx

∫

T ∗expxv

∫

Texpxv

e−i(〈v,ξ〉+〈w,η〉)a(x, ξ)(Jexpxv

x b(expxv, η))J(x; v, w)u(expexpxvw) dwdηdvdξ.

(8.14)We transform integral (8.14) along with the same line as has been used for treating

integral (5.2), and obtain the following analogs of formulas (5.11)–(5.12):

ABu(x) = (2π)−n

∫

T ∗x X

∫

TxX

e−i〈v,ξ〉d(x, v, ξ)Jexpxvx u(expxv) dvdξ, (8.15)

(2π)nd(x, v, ξ) =

=

∫

T ∗x X

∫

TxX

eiψ(x,v,ξ;w,η)a(x, η)(Jexpxw

x b(expxw, tF (x, v, w)ξ))J(x; w, exp−1

expxw expxv) dwdη,

(8.16)where F and ψ are the same as in (5.12). Applying Lemma 8.1, we obtain the geometricsymbol c(x, ξ) of the product C = AB in the form

(2π)2nc(x, ξ) =

=

∫ ∫ ∫ ∫eiχ(x,ξ;v,η,w,ζ)a(x, ζ)

(Jexpxw

x b(expx w, tF (x, v, w)η))J ′(x; v, w) dwdζdηdv,

where χ is the same as in (5.13) and

J ′(x; v, w) = J(x; w, exp−1expxw expxv).

Transforming the latter integral in the same way as (5.13) has been treated, we obtainthe following analog of (5.15):

(λ

2π

)−2n

c(x, λξ) =

=

∫ ∫ ∫ ∫eiλϕ(x,ξ;v,η,w,ζ)a(x, λη)

(Jexpxv

x b(expxv, λtH(x, v, w)ζ))J ′′(x; v, w)dvdηdwdζ,

(8.17)where ϕ,H are the same as in (5.15), and

J ′′(x; v, w) = J ′(x; v + w, v).

Applying the stationary phase method to integral (8.17) in the same way as in Section5, we obtain the following analog of formula (5.44):

c(x, ξ) ∼∑

α

1

α!

v

∇αa(x, ξ)×

55

×Dαv

[∑

β

1

β!∂β

ξ Dβw

(ei〈h(x;v,w),ξ〉(Jexpxv

x b(expxv, tH(x; v, w)ξ))J ′′(x; v, w))∣∣

w=0

]

v=0

.

(8.18)The operator J

expxvx is independent of w and commutes with Dβ

w. Therefore the equal-ity

Dβw

(ei〈h(x;v,w),ξ〉(Jexpxv

x b(expxv, tH(x; v, w)ξ))J ′′(x; v, w))∣∣

w=0

= Jexpxvx

[∑

λ≤β

Dλw

(ei〈h(x;v,w),ξ〉(b(expxv, tH(x; v, w)ξ))

)∣∣w=0

]· P β

λ (x, v)

holds with some smooth End (Ex)-valued functions P βλ (x, v) which are expressed through

partial derivatives of the function J ′′(x; v, w) with respect to w at w = 0. Next, trans-forming the expression in brackets in the same way as in Section 5, we obtain the followinganalog of (5.46):

Dβw

(ei〈h(x;v,w),ξ〉(Jexpxv

x b(expxv, tH(x; v, w)ξ))J ′′(x; v, w))∣∣

w=0

=∑

|γ|≤|β|

(Jexpxv

x

v

∇γb(expxv, t(exp′x(v))−1ξ))· qβ

γ (x, v, ξ), (8.19)

where qβγ (x, v, ξ) are operator-valued polynomials of degree ≤ |β|/2 + |γ| in ξ.

Next, we differentiate (8.19) with respect to ξ and use the fact that Jexpxvx commutes

with ∂ξ. In such the way we obtain the following version of (5.47):

∂βξ Dβ

w

(ei〈h(x;v,w),ξ〉(Jexpxv

x b(expxv, tH(x; v, w)ξ))J ′′(x; v, w))∣∣

w=0

=∑

|β|/2≤|γ|≤2|β|Jexpxv

x

( v

∇γb(expxv, t(exp′x(v))−1ξ))· pβ

γ(x, v, ξ), (8.20)

where pβγ(x, v, ξ) are operator-valued polynomials of degree ≤ |γ|− |β|/2. Then we obtain

the following analog of (5.49):

Dαv

[∑

β

1

β!∂β

ξ Dβw

(ei〈h(x;v,w),ξ〉(Jexpxv

x b(expxv, tH(x; v, w)ξ))J ′′(x; v, w))]

v=w=0

=∑

β≤α

∑γ

Dβv

[Jexpxv

x

( v

∇γb(expxv, t(exp′x(v))−1ξ))]

v=0· qα

βγ(x, ξ), (8.21)

where qαβγ(x, ξ) are some operator-valued polynomials of degree ≤ 3|γ|/4.

Validity of the representation

Dβv

[Jexpxv

x

( v

∇γb(expxv, t(exp′x(v))−1ξ))]

v=0=

∑

λ≤β

∑

|γ|≤|µ|≤|β−λ+γ|(−i

h

∇)λv

∇µb(x, ξ)·pβγλµ(x, ξ)

(8.22)with deg pβγ

λµ ≤ |µ|− |γ| is proved in the same way as (5.51)–(5.52) have been proven withthe only exception: we use (7.20) instead of (1.20).

56

Substituting expression (8.22) into (8.21), we obtain the following version of (5.58):

Dαv

[∑

β

1

β!∂β

ξ Dβw

(ei〈h(x;v,w),ξ〉(Jexpxv

x b(expxv, tH(x; v, w)ξ))J ′′(x; v, w))]

v=w=0

=∑

β≤α

∑γ

(−ih

∇)βv

∇γb(x, ξ) · pαβγ(x, ξ), (8.23)

where pαβγ(x, ξ) are operator-valued polynomials of degree ≤ (|α| − |β|)/4 + 3|γ|/4 in ξ.

Finally, substituting (8.23) into (8.18), we obtain (5.59)–(5.60). The proof is nowfinished with the help of Theorem 7.5 by the same arguments as in Section 5. The remarkat the end of Section 5 is still valid.

Let us generalize Theorem 6.1 to the case of vector bundles. For a vector bundle E,let E∗ be the antidual bundle whose fiber over x ∈ X is the antidual space E∗

x of Ex. Forw ∈ C∞(E∗), the section w ∈ C∞(E∗) is well defined.

Given a vector bundle E over X and a real κ, there is the natural L2-pairing betweenE-valued κ-densities and E∗-valued (1− κ)-densities

(u,w)L2 =

∫

X

〈u(x), w(x)〉 for u ∈ C∞(E ⊗ ΩκX) and w ∈ C∞(E∗ ⊗ Ω1−κX).

(8.24)Here either u or w is assumed to be compactly supported, and 〈·, ·〉 is the pairing betweenE and E∗. We restrict ourselves to considering operators on half-densities, i.e., to thecase of κ = 1/2. The reader can easily generalize the result to the case of an arbitrary κ.

Theorem 8.4 Let (X,∇) be a manifold with connection, (E,∇E) and (F,∇F ) be twovector bundles with connection over X. For an operator A ∈ Ψm(X,∇; E ⊗ Ω1/2X,F ⊗Ω1/2X), the dual operator

A∗ : C∞0 (F ∗ ⊗ Ω1/2X) → C∞(E∗ ⊗ Ω1/2X)

with respect to the L2-pairing (8.24) belongs to Ψm(X,∇; F ∗ ⊗ Ω1/2X, E∗ ⊗ Ω1/2X) andhas the geometric symbol b(x, ξ) which is expressed through the geometric symbol a(x, ξ)of A by the asymptotic series

b(x, ξ) ∼∑

α

1

α!(−i

h

∇)αv

∇αa∗(x, ξ),

where a∗(x, ξ) = ta(x, ξ) : F ∗x → E∗

x is the antidual operator of a(x, ξ) : Ex → Fx.

Note that the operators A and A∗ are independent of the connection ∇F but thegeometric symbol of A∗ depends on the latter connection trough the horizontal derivativesh

∇αa∗(x, ξ).We omit the proof of this theorem which goes along with the same line as the proof

of Theorem 6.1. For instance, formula (6.2) will be replaced with the following one:

A∗u(x) = (2π)−n

∫

T ∗x X

∫

TxX

e−i〈v,ξ〉(Jexpxv

x a(expxv, tIexpxvx ξ)

)Jexpxv

x u(expxv) dvdξ.

We have to use (7.14) instead of (2.20) at the final step of the proof.

57

References

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[2] Eisenhart L. F. Riemannian Geometry. Princeton University Press, 1926.

[3] Hormander L. Analysis of Linear Partial Differential Operators I. DistributionTheory and Fourier Analysis. Springer–Verlag, Berlin Heidelberg New York Tokio,1983.

[4] Hormander L. Analysis of Linear Partial Differential Operators III. Pseudo-Differential Operators. Springer–Verlag, Berlin Heidelberg New York Tokio, 1985.

[5] Pflaum M. J. The normal symbol on Riemannian manifolds. New York J. Math.,1998, 4, 97–125.

[6] Sharafutdinov V. A. Integral Geometry of Tensor Fields. VSP, Utrecht, the Nether-lands, 1994.

[7] Widom H. Families of pseudodifferential operators. Topics in Functional Analysis(I. Gohberg and M. Kac, eds.), 1978, Academic Press, New York, 345-395.

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