differential invariants of riemannian and einstein...

Differential invariants of Riemannianand Einstein manifolds

Alexei Kotov(joint work with Valentin Lychagin)

Institute of Mathematics and StatisticsUniversity of Tromso

Nonlinear Mathematical Physics: Twenty Years of JNMPJune, 4-14, 2013

Alexei Kotov (joint work with Valentin Lychagin) Differential invariants of Riemannian and Einstein manifolds

Riemannian metrics on a manifold

Let M be a smooth n−dimensional manifold M.

Definition A Riemannian (pseudo-Riemannian) structure onM is a smooth section g ∈ Γ(S2T ∗M) which is non-degenerateand positive (indefinite) at each point x ∈ M.

Let (M, g) be a (pseudo-)Riemannian manifold. Then there existsa unique connection ∇ in TM (the Levi-Civita connection)which is

1. torsion-free:

∇X (Y )−∇Y (X )− [X ,Y ] = 0, ∀X ,Y ∈ Γ(TM)

2. metric preserving:

∇X (g) = 0 ,∀X ∈ Γ(TM)

1. torsion-free:

∇X (Y )−∇Y (X )− [X ,Y ] = 0, ∀X ,Y ∈ Γ(TM)

∇X (g) = 0 ,∀X ∈ Γ(TM)

1. torsion-free:

∇X (Y )−∇Y (X )− [X ,Y ] = 0, ∀X ,Y ∈ Γ(TM)

∇X (g) = 0 ,∀X ∈ Γ(TM)

Riemannian metric and Levi-Civita connection incoordinates

Let x i , i = 1, . . . , n be local coordinates on M. Then g representsas follows:

2gijdx

The Christoffel symbols of ∇ with respect to x i are:

Γkij = gkl (∂igjl + ∂jgil − ∂lgij)

where (gkl) is the inverse matrix and ∂i is the partial derivativealong x i .

Riemannian curvature tensor

Definition The Riemannian curvature tensor R of (M, g) is thecurvature tensor of the Levi-Civita connection

R(X1,X2)X3 =([∇X1 ,∇X2 ]−∇[X1,X2]

We identify R with a (0, 4)−tensor: for all X1,X2,X3,X4 ∈ Γ(TM)

R(X1,X2,X3,X4) = g (R(X1,X2)X3,X4)

The Riemannian curvature tensor is skew-symmetric with respectto the first and second couple of indices:

R(X1,X2,X3,X4) = −R(X2,X1,X3,X4) = −R(X1,X2,X4,X3)

and satisfies the algebraic Bianchi identity

R(X1,X2,X3,X4) + R(X2,X3,X1,X4) + R(X3,X1,X2,X4) = 0

Riemannian curvature tensor

Definition The Riemannian curvature tensor R of (M, g) is thecurvature tensor of the Levi-Civita connection

R(X1,X2)X3 =([∇X1 ,∇X2 ]−∇[X1,X2]

We identify R with a (0, 4)−tensor: for all X1,X2,X3,X4 ∈ Γ(TM)

R(X1,X2,X3,X4) = g (R(X1,X2)X3,X4)

The Riemannian curvature tensor is skew-symmetric with respectto the first and second couple of indices:

R(X1,X2,X3,X4) = −R(X2,X1,X3,X4) = −R(X1,X2,X4,X3)

and satisfies the algebraic Bianchi identity

R(X1,X2,X3,X4) + R(X2,X3,X1,X4) + R(X3,X1,X2,X4) = 0

The last two properties combined together imply that

R(X1,X2,X3,X4) = R(X3,X4,X1,X2)

Equivalently, R is a section of S2(Λ2T ∗M) which belongs to thekernel of the wedge product

S2(Λ2T ∗M)→ Λ4T ∗M

The second Bianchi identity tells us that the covariant derivativeof R with respect to ∇, being a section of T ∗ ⊗ S2(Λ2T ∗M),belongs to the kernel of the natural map

T ∗⊗S2(Λ2T ∗M) ⊂(T ∗ ⊗ Λ2T ∗M

)⊗Λ2T ∗M → Λ3T ∗M⊗Λ2T ∗M

determined by the exterior multiplication T ∗ ⊗ Λ2T ∗M → Λ3T ∗M.

The last two properties combined together imply that

R(X1,X2,X3,X4) = R(X3,X4,X1,X2)

Equivalently, R is a section of S2(Λ2T ∗M) which belongs to thekernel of the wedge product

S2(Λ2T ∗M)→ Λ4T ∗M

The second Bianchi identity tells us that the covariant derivativeof R with respect to ∇, being a section of T ∗ ⊗ S2(Λ2T ∗M),belongs to the kernel of the natural map

T ∗⊗S2(Λ2T ∗M) ⊂(T ∗ ⊗ Λ2T ∗M

)⊗Λ2T ∗M → Λ3T ∗M⊗Λ2T ∗M

determined by the exterior multiplication T ∗ ⊗ Λ2T ∗M → Λ3T ∗M.

Ricci curvature tensor

Definition The Ricci curvature tensor Ric of (M, g) is the thetrace of the operator Y 7→ R(Y ,X1)X2.

I Thanks to the algebraic properties of R, the Ricci tensor issymmetric.

I We can identify it with a symmetric operator Ric such that

Ric(X1,X2) = g(Ric(X1),X2

Definition The scalar curvature S of (M, g) is the the trace of

Ricci curvature tensor

Definition The Ricci curvature tensor Ric of (M, g) is the thetrace of the operator Y 7→ R(Y ,X1)X2.

I Thanks to the algebraic properties of R, the Ricci tensor issymmetric.

I We can identify it with a symmetric operator Ric such that

Ric(X1,X2) = g(Ric(X1),X2

Definition The scalar curvature S of (M, g) is the the trace of

The space of Riemannian curvature tensors at x ∈ M is anO(Tx)−module, where O(Tx) is the group of isometries of(TxM, gx).

The Riemannian curvature tensor R admits thefollowing irreducible decomposition:

R = W +1

n − 2(Ric − 1

ng) ? g +

2n(n − 1)g ? g

where ? is the Kulkarni-Nomizu product defined for the coupleof symmetric (0, 2)−tensors h1 and h2 as follows:

h1 ? h2(X1,X2,X3,X4) =

= h1(X1,X3)h2(X2,X4) + h1(X2,X4)h2(X1,X3)−−h1(X1,X4)h2(X2,X3) − h1(X2,X3)h2(X1,X4)

Definition W is called the Weyl tensor of (M, g).

The space of Riemannian curvature tensors at x ∈ M is anO(Tx)−module, where O(Tx) is the group of isometries of(TxM, gx). The Riemannian curvature tensor R admits thefollowing irreducible decomposition:

R = W +1

n − 2(Ric − 1

ng) ? g +

2n(n − 1)g ? g

h1 ? h2(X1,X2,X3,X4) =

The space of Riemannian curvature tensors at x ∈ M is anO(Tx)−module, where O(Tx) is the group of isometries of(TxM, gx). The Riemannian curvature tensor R admits thefollowing irreducible decomposition:

R = W +1

n − 2(Ric − 1

ng) ? g +

2n(n − 1)g ? g

h1 ? h2(X1,X2,X3,X4) =

Equivalence problem

Classification of metrics up to (local) diffeomorphisms.

Let π : E (s)→ M be the bundle of non-degenerate metrics ofsignature (n − s, s): E (s) ⊂ S2T ∗M.

The main goal is to describe the quiotient the space of (local)sections of π by the action of the (pseudo-)group of (local)diffeomorphisms.

A simpler problem (formal classification of metrics) is todescribe the quotient of the space of infinite jets of metrics by theaction of D, the (pseudo-)group of local diffeomorphisms.

Equivalence problem

Jet bundles

Let π : E → M be a bundle on M of dimension n andπk : Jkπ → M be the bundle of k−jets of local sections of π.

Jkπ is (an affine) bundle over J lπ for all k ≥ l , such that πk,l isan inverse system, i.e.

πl ,r πk,l = πk,l

for all k ≥ l ≥ r .

Definition J∞π = lim←− Jkπ.

There exists a canonical vector bundle τk+1 of rank n over Jk+1π,k ≥ 0, such that the fiber of τk+1 at any xk+1 = [s]k+1

x coincideswith the tangent subspace to the k−jet of s at xk = [s]kx . Here s isany local section of π at x ∈ M, representing xk+1 ∈ Jk+1π.

Jet bundles

I τk+1 is subbundle of π∗k+1,k

(TJkπ

), so that sections of τk+1

are (particular) derivations of functions on Jkπ with values infunctions on Jk+1π;

I Let Cxk the linear span of τk+1 at all xk+1 over xk , consideredas subspaces of TxkJ

kπ. Then C = ∪xkCxk is the Cartandistribution on Jkπ, such that k−jets of sections of π aren−simensional integral (and projectible) submanifolds of C;

I τk → Jkπ is an inverse system; its projective limit τ is asubbundle of TJ∞π;

I Sections of τ which are projectible onto M are totalderivatives.

I The algebra of those functions which are algebraic on jetfibers, is generated as (a differential algebra) by totalderivatives and functions on the total space of π.

k−jets of local metrics

Let πk : Jkπ → M be the bundle of k−jets of local metrics. Thepseudogroup of diffeomorphisms D acts on Jkπ as follows: if Φ isa local diffeomorphism, such that Φ(x) = y , and [g ]ky is the k−jet

of a metric g at y , then Φ∗([g ]ky

)= [Φ∗(g)]kx .

The action of D commutes with the projections πk,l , thus D alsoacts on the projective limit - the infinite jet bundle.

Let us define Mk(s) = Jkπ/D. Given that the action of D on Mis transitive, one has

Mk(s) ' Jkπx/Gkx

where Jkπx is the fiber of the bundle of k−jets over a point x ∈ Mand G k

x is the isotropy group of x : Gx coincides with the group ofk−jets of diffeomorphisms which preserve x .

)= [Φ∗(g)]kx .

Mk(s) ' Jkπx/Gkx

)= [Φ∗(g)]kx .

Mk(s) ' Jkπx/Gkx

Geodesic coordinates

Let us fix x ∈ M and take an orthogonal frame in TxM withrespect to g . Then there exists a unique system of coordinates x i

with the center in xsuch that straight lines passing through theorigin - (tx1, . . . , txn), t ≥ 0 - are geodesics with respect to g (toavoid the problem of convergence, formal calculus is sufficient).

Taylor coefficients of g in geodesic coordinates

The classical result asserts that ∀k ≥ 2 one has

∂l1 . . . ∂lpgij = Constant (Ril1jl2;l3...lk + Q) x l1 . . . x lk

where Ril1jl2;l3...lk are the coordinate components of thesymmetrized k−th order covariant derivative of the curvaturetensor R (the latter is a section of

⊕p≤k S

pT ∗M ⊗ S2(Λ2T ∗M))and Q is a polynomial in lower order covariant derivatives of R.

Geodesic coordinates

Let us fix x ∈ M and take an orthogonal frame in TxM withrespect to g . Then there exists a unique system of coordinates x i

with the center in xsuch that straight lines passing through theorigin - (tx1, . . . , txn), t ≥ 0 - are geodesics with respect to g (toavoid the problem of convergence, formal calculus is sufficient).

Taylor coefficients of g in geodesic coordinates

The classical result asserts that ∀k ≥ 2 one has

∂l1 . . . ∂lpgij = Constant (Ril1jl2;l3...lk + Q) x l1 . . . x lk

where Ril1jl2;l3...lk are the coordinate components of thesymmetrized k−th order covariant derivative of the curvaturetensor R (the latter is a section of

⊕p≤k S

pT ∗M ⊗ S2(Λ2T ∗M))and Q is a polynomial in lower order covariant derivatives of R.

As a corollary of the isomorphism between formal neighborhood ofx and TxM, which is determined by geodesic coordinates, andtaking into account that the Taylor coefficients in geodesiccoordinates of g at x are in one-to-one correspondence with thecomponents of the symmetrized up to k−th order covariantderivative of R, we obtain:

Mk(s) ' Rkx/O(Tx)

where Rkx is the space of symmetrized up to k−th order covariant

derivatives of curvatures at x .

Taking into account that the quotient space is independent on thechoice of x , hereafter we omit the subscript x .

As a corollary of the isomorphism between formal neighborhood ofx and TxM, which is determined by geodesic coordinates, andtaking into account that the Taylor coefficients in geodesiccoordinates of g at x are in one-to-one correspondence with thecomponents of the symmetrized up to k−th order covariantderivative of R, we obtain:

Mk(s) ' Rkx/O(Tx)

where Rkx is the space of symmetrized up to k−th order covariant

derivatives of curvatures at x .

Taking into account that the quotient space is independent on thechoice of x , hereafter we omit the subscript x .

Differential invariants

Definition Functions on Jkπ, which are invariant under the actionof a given pseudogroup (in this case - the pseudogroup of localdiffeomorphisms), are called differential invariants of order k.

In the case of metrics we see that differential invariants are inone-to-one correspondence with the algebraic invariants of thealgebraic quotient of Rk by the action of the orthogonal group.The classical theorem tells us that the algebra of such invariants isgenerated by polynomials which are obtained by pairing in allpossible ways by use of metric.

Such a description is rather explicit. However, it is hard tocompute the relations between generators. Later we shall faceanother difficulties.

From formal to local equivalence problem

For any (pseudo-)Riemannian manifold (M, g) we have a naturalmap M →Mk for all k given by the evaluation of all differentialinvariants.

The following questions appear:

I What is the property of the image of (M, g)? Can it bearbitrary?

I Let us assume that the images of two (pseudo-)Riemannianmanifolds (M, g) and (M ′, g ′) coincide: can we make aconclusion that (M, g) and (M ′, g ′) are equivalent?

The pure algebraic description is, strictly speaking, not an answerto any of those questions. Moreover, the number of algebraicgenerators of all differential invariants is obviously infinite. Thisimplies that, naively, we should check infinitely many conditions.

Differential approach

Now is the time to remember that the algebra of functions on jetsis a differential algebra. One can expect a similar behavior ofdifferential invariants.

Invariant derivations

The pseudogroup of diffeomorphisms D acts on τ , such thatτ → J∞π is an equivariant bundle. We take the quotient:

L := τ/D

L is a bundle on M in the following sense: sections of L, whichare D−invariant derivatives, makes a module over the algebra ofdifferential invariants.

Let M∞ = lim←−Mk . It turns out that the ideal of functions

vanishing on the image of (M, g) in M∞ is a differential ideal - itis closed under the action of L. The latter is a counterpart of theproperty that jets of sections are integral submanifolds of theCartan distribution. Although there is no Cartan destributionanymore, the action of invariant derivatives on differentialinvariants is the ”shadow of that”.

Invariant derivations

The pseudogroup of diffeomorphisms D acts on τ , such thatτ → J∞π is an equivariant bundle. We take the quotient:

L := τ/D

L is a bundle on M in the following sense: sections of L, whichare D−invariant derivatives, makes a module over the algebra ofdifferential invariants.

Let M∞ = lim←−Mk . It turns out that the ideal of functions

vanishing on the image of (M, g) in M∞ is a differential ideal - itis closed under the action of L. The latter is a counterpart of theproperty that jets of sections are integral submanifolds of theCartan distribution. Although there is no Cartan destributionanymore, the action of invariant derivatives on differentialinvariants is the ”shadow of that”.

Lie-Tresse theorem

The recent version of the Lie-Tresse theorem (Kruglikov-Lychagin,2011) asserts that the algebra of rational differential invariants isfinitely generated as a differential algebra. This means that thereexists a finite number of rational invariants and rational invariantderivatives which generate all rational differential invariants.

Lie-Tresse theorem and the equivalence problemfor metrics

Assume that the image of (M, g) belongs to the regular strata inM∞. Then

I The requirement to be a differential ideal means that theimage of (M, g) is a solution to a certain differential equation;

I The images of two (pseudo-)Riemannian manifolds (M, g)and (M ′, g ′) coincide if and only if they are (at least locally)diffeomorphic;

I Regarding the last statement, there is a finite number ofconditions to be verified. These conditions can be explicitlycomputed.

I Non-regular cases must be treated independently. However,some non-regular cases (eg. Einstein metrics) admit a similardescription with its own notion of regularity.

Example 1: 3−dimensional Riemannian manifold

Given that S2(Λ2TM) ' S2TM, the curvature tensor R is uniquelydetermined by its Ricci part Ric .

The regularity condition

I On the level of 2−d order jets: Ric has mutually differenteigenvalues;

I On the level of 3−d order jets: the following differential

invariants I1 := S , I2 = Tr(Ric

, and I3 = Tr(Ric

functionally independent, i.e. dI1 ∧ dI2 ∧ dI3 6= 0.

Basic invariants and derivatives

I Basic invariant derivatives: the frame Xi dual to dIi,i = 1, 2, 3;

I Basic invariants: the coefficients of g in coordinates Ii , thatis, g(Xi ,Xj).

, and I3 = Tr(Ric

The regular part in the quotient space is the infinitely prolongeddifferential equation:

Tr(Ric(g)i

This case can be easily generalized to arbitrary dimensions. We callthis regularity condition the Ricci regularity.

However, in higher dimensions the regularity condition can bechosen in other ways - thanks to the Weyl tensor.

The regular part in the quotient space is the infinitely prolongeddifferential equation:

Tr(Ric(g)i

This case can be easily generalized to arbitrary dimensions. We callthis regularity condition the Ricci regularity.

However, in higher dimensions the regularity condition can bechosen in other ways - thanks to the Weyl tensor.

Example 2: 4−dimensional Einstein manifold,Lorentz signature

In this case Ric = const ∗ g , therefore the curvature tensor R isuniquely determined by its Weyl part W .

We fix a volume form µq. The corresponding Hodge operator ∗,determined by the condition

g−1(α, ∗β)µg = α ∧ β , α ∈ Ωq(M), β ∈ Ωn−q(M)

will satisfy the property ∗2 = −1 on 2−forms. That makes Λ2T ∗Minto a complex bundle and, given that W , being considered as anoperator W acting in 2−forms, commutes with ∗, also the Weyltensor into a complex symmetric operator. Moreover, its complextrace is 0 because of the algebraic Bianchi identity.

The regularity condition and basic invariantsfor an 4−dimensional Einstein manifold withLorentz signature

I On the level of 2−d order jets: W has mutually differentcomplex eigenvalues (corresponds to the first Petrov’s type);

I On the level of 3−d order jets: the real and imaginary parts of

and TrC

are functionally independent;

I The choice of basic invariants, derivatives and the descriptionof the regular strata of the space of invariants as a differentialequation are evident (similar to the Ricci case).

differential invariants of riemannian and einstein...

Documents