Almost analytic Kahler forms with respect to a quadratic

endomorphism with applications in Riemann-Finsler

geometry

Mircea Crasmareanu and Laurian-Ioan Piscoran

December 5, 2017; Revision: March 19, 2018

Abstract

The almost analyticity with respect to a quadratic endomorphism T is introduced in analgebraic setting concerning a commutative and associative algebra A. Two main propertiesare proved: the first concerns the simultaneous closedness for an almost analytic 1-form ωand Tω while the second regards the vanishing of the interior product of such a form with theNijenhuis tensor of T . Also, we introduce an extension of the Frolicher-Nijenhuis formalismto this framework as well as a hermitian type property. When A is the algebra of smoothfunctions on a given (even dimensional) manifold we recover the classical notion of almostanalytic 1-form. We study this analyticity and the hermitian type property for the Cartan1-form of a Riemann-Finsler geometry. Also, we study the almost analytic functions on thetangent bundle of a Riemann-Finsler geometry with respect to the associated almost para-complex and almost complex structure of this geometry. We introduce two new types ofHessian and respectively Laplacian corresponding to these structures. Two types of gradientRicci solitons are introduced in the tangent bundle.

2010 Mathematics Subject Classification: 53C15; 53A10; 16W20; 16W25; 15A75; 53C60.Key Words: quadratic endomorphism; almost T -analytic Kahler form; Nijenhuis tensor;T -hermitian form; ddT -Lemma; Riemann-Finsler geometry; Laplacian; gradient Ricci soliton.

Introduction

The notion of almost analytic form was introduced a long time ago in the almost complex geometryand hence, it was treated in local coordinates, especially by Japanese geometers [25]. A globalapproach appeared in [22], unfortunately only in Romanian. Some of these global techniques wereused in [20] and [21]; for example in the former paper a differential is introduced in the algebra ofalmost analytic forms and a corresponding Poincare type lemma is proved.

In a recent work, namely [14], we propose a unifying setting of almost analyticity of formswhich include both (almost) complex and (almost) para-complex structure. More precisely, givena quadratic endomorphism F on a smooth (even dimensional) manifold M let us consider its dualtensor F ∗ on differential forms. Then an almost F -analytic 1-form ω on M is closed if and only if

1

its conjugate F ∗ω is closed. Also, we extend this notion to arbitrary r-forms with r lower or equalhalf the dimension of M and in the paper [13] we extend all this techniques on Lie algebroids.

The aim of present note is to generalize this differential geometric topic into an algebraicframework using the setting of Kahler forms provided by Chapter 3 of [18]. More precisely,a (differential) 1-form ω is replaced here with an element from a commutative and associativealgebra A (over a field F) and a (smooth) vector field X is replaced with a derivation of A.Having an (quadratic) endomorphism T of A we associate a conjugate of X with respect to T anddefine ω as being almost analytic with respect to T through a relation involving the differentialdω. A main feature of such almost analytic forms is that ω and Tω are simultaneous closed ornot, conform Proposition 1.2. Moreover, this result holds also with the differential dT induced byT through the Frolicher-Nijenhuis type calculus. Another important property of almost analytic1-forms which extends in this algebraic setting is the vanishing of the interior product of ω withthe conjugate of Nijenhuis tensor of T ; see Proposition 1.4. We add also to our study a property ofhermitian type in this setting. The first section ends with the expression of the d2

T which vanisheswhen applied to almost T -analytic elements of A. In the general case, d2

T admits a simplificationif we add a ddT -Lemma.

As application of this study we consider the Riemann-Finsler geometry (N,F ) which is na-turally endowed with an almost complex structure ΨF , an almost para-complex one PF and a1-form ωF . We obtain that this 1-form behaves similar for PF and ΨF : is not almost analytic butis hermitian. We study also the smooth functions on T0N which are (weak) almost analytic withrespect to PF and ΨF . More precisely, in the para-complex case we obtain, in particular, all thesmooth functions on the base N while for the complex case, we derive a very complicated equation.In the particular case of a Riemannian geometry and a search restricted to the base functions forthe almost complex situation we obtain the totally geodesic functions i.e. the functions withvanishing Hessian.

This example yields the introduction of two new types of Hessian and Laplacian in Riemann-Finsler geometry; hence an almost analytic function with respect to PF is almost para-harmonicand an almost analytic function with respect to ΨF is an almost complex-harmonic. On thisway we extend the usual implication from the theory of complex functions: analyticity impliesharmonicity. We give the value of these Laplacians on the energy F 2 as well as the expression onEuclidean and Berwald geometries. A global expression is obtained for the almost para-complexLaplacian in a Landsberg geometry by using the vertical gradient and the divergence with respectto the Cartan or Chern-Rund connections. Due to these Hessians we introduce two new types ofgradient Ricci solitons in the tangent bundle of a Riemann-Finsler geometry (N,F ) by means ofthe Jacobi endomorphism of (N,F ). We point out that recently, the Ricci solitons are used in [10]to express the Zermelo navaigation problem for Randers metrics of constant flag curvature c andthe class of our gradient Ricci solitons provided by Proposition 3.8 concerns exactly with this caseof constant flag curvature.

The last section concerns with the study of Hessian and Laplacian in the tangent bundle of aRiemann-Finsler geometry. Since in literature appear already several ”horizontal” and ”vertical”Laplacians, see [17] and [27], we discuss both the Hessian and Laplacian associated to the Sasakimetric GF respectively the Hessian and Laplacian generated by a Finslerian connection; note thatthe Levi-Civita connection of the Riemannian metric GF is not a Finslerian one. As main toolswe deal with the Cartan, Chern-Rund, Berwald and Hashiguchi connection and it follows that thealmost-complex Laplacian introduced in the previous section is exactly the Berwald Laplacian.

2

1 The algebraic approach of almost analyticity

Let F be a field of characteristic zero and A a commutative and associative F-algebra. After [18, p.69], the module of Kahler differentials of A is the A-module Ω1(A), which is the free A-modulegenerated by the set d(F )|F ∈ A, modulo the submodule generated by all elements of eitherone of the following three types: i) d(F +G)−d(F )−d(G), ii) d(FG)−Fd(G)−G(dF ), iii) d(a),where a ∈ F and F,G ∈ A. It results the map d : A → Ω1(A) given by d : F → d(F ).

Now we consider the A-module Ωp(A) = ∧pΩ1(A) where ∧ is the wedge product over A andΩ0(A) := A. Using the expression of [18, p. 70], the elements of Ωp(A) with p > 0 are called Kahlerp-forms. The F-linear map d extends by functoriality of ∧ to an F-linear map ∧•d : ∧•A → Ω•(A):

∧•d(F1 ∧ ... ∧ Fp) := d(F1) ∧ ... ∧ d(Fp) (1.1)

for any F1, ...., Fp ∈ A. Also, d extends to a graded F-map d : Ω•(A)→ Ω•+1(A) called de Rhamdifferential and given by:

d(GdF1 ∧ ... ∧ dFp) := dG ∧ dF1 ∧ .... ∧ dFp (1.2)

for all G,F1, ..., Fp ∈ A. It is a graded derivation of degree 1 of (Ω•(A),∧):

d(ω ∧ η) = dω ∧ η + (−1)pω ∧ dη (1.3)

for ω ∈ Ωp(A) respectively η ∈ Ω•(A). As usual, d d = 0 and if dω = 0 then ω is called closedand if ω = dθ then ω is called exact, [18, p. 72].

The next A-module we need is the set X 1(A) of derivations of A given by [18, p. 69]:

X 1(A) = HomA(Ω1(A),A). (1.4)

It results that any X ∈ X 1(A) can be considered as X : A → A with:

X(F ) := X(dF ) ∈ A (1.5)

for every F ∈ A; but for the simplification of notations we give up to use the hat. From thedefinition of Ω1(A) it results:

X(FG) = X(d(FG)) = X(FdG+GdF ) = FX(dG) +GX(dF ) = FX(G) +GX(F ) (1.6)

which means that X is indeed a derivation of A. Also, every X ∈ X 1(A) yields a graded A-linearmap of degree −1:

iX : Ω•(A)→ Ω•−1(A) (1.7)

the express of it can be find in [18, p. 76]. Note that we have iX iX = 0 and if ω ∈ Ω1(A) theniXω = X(ω).

Fix now a F-linear map T : Ω1(A) → Ω1(A) satisfying the nilpotence T 2 = ε1Ω1(A) where

ε = ±1 ∈ F. Then to any derivation X ∈ X 1(A) we associate its T -conjugate X ∈ X 1(A):

X(ω) :=1

εX(Tω) = εX T (ω) (1.8)

for all ω ∈ Ω1(A). From X = εX T it results X = εX.

3

Definition 1.1 ω ∈ Ω1(A) is called almost T -analytic if:

iXdω = εiXd(Tω) (1.9)

for all X ∈ X 1(A). Let Ω1(A, T ) be the F-module of these forms.

The first main properties of almost T -analytic Kahler forms are provided by:

Proposition 1.2 i) If ω ∈ Ω1(A, T ) then ω is closed if and only if Tω is closed.ii) Let ω ∈ Ω1(A). Then ω ∈ Ω1(A, T ) if and only if Tω ∈ Ω1(A, T ).

Proof i) and ii) are direct consequences of (1.9) and the similar equation with X → X:

iXdω = iXd(Tω). (1.10)

(1.9) means that: Ω1(A, T ) = ∩X∈X 1(A)KeriX d − εiX d T = ∩X∈X 1(A)KeriXT d −iX d T. 2

Let us remark that the definition means that for all Y ∈ X 1(A) we have:

Y (iXdω) = εY (iXd(Tω)). (1.11)

Another main tool is the Lie derivative on X 1(A) given by the usual bracket of derivatives:

LXY (F ) := X(Y (F ))− Y (X(F )) (1.12)

for all F ∈ A and X,Y ∈ X 1(A). Hence a direct computation gives the general formula for1-forms:

Y (iXdω) = X(iY ω)− Y (iXω)− iLXY ω. (1.13)

Remarks 1.3 i) The general formula of differential d is equation (3.28) of [18, p. 75].ii) From (1.13) it follows that ω ∈ Ω1(A) is closed if and only if:

X(iY ω)− Y (iXω) = iLXY ω (1.14)

for any X,Y ∈ X 1(A). 2

We have that (1.11) means:

X(iY ω)− Y (iXω)− iLXY ω = ε [X(iY Tω)− Y (iXTω)− iLXY Tω] . (1.15)

Since (1.8) means:iXTω = εiXω (1.16)

this relation can be written also as:

X(iY ω)− Y (iXω)− iLXY ω = X(iY ω)− Y (iXω)− iLXY ω (1.17)

equivalently:X(iY ω)− iLXY ω = X(iY ω)− iLXY ω. (1.18)

Due to the Lie derivative we introduce the Nijenhuis tensor of T as NT : X 1(A) × X 1(A) →X 1(A) given by:

NT (X,Y ) := LXY − LXY − LXY + εLXY (1.19)

4

which gives a third property of almost T -analytic Kahler forms:

Proposition 1.4 If ω ∈ Ω1(A, T ) then:

iNT ω = 0(= iNT Tω). (1.20)

Proof We must prove that:

iLXY−εLXY−εLXY+εLXYω = 0 (1.21)

for any X,Y ∈ X 1(A). But (1.18) means:

iLXY−LXYω = X(iY ω)−X(iY ω). (1.22)

With X → X and Y → Y this relation becomes:

iLXY−εLXYω = εX(iY ω)− εX(iY ω). (1.23)

Then, the sum of (1.23) with the multiplication of (1.22) with ε yields (1.21). From iNT ω =εNT T (ω) and the first part it result the second part of (1.20). 2

It is natural to consider the cases below:

Definition 1.5 i) T is called integrable if NT = 0. ii) ω ∈ Ω1(A) is called T -analytic ifω ∈ Ω1(A, T ) and T is integrable.

Hence ω is T -analytic if and only if Tω is so. In order to provide examples of elements fromΩ1(A, T ) we introduce:

Definition 1.6 F ∈ A is called almost T -analytic if there exists G ∈ A such that the almostε-CR equation holds:

T (dF ) = dG. (1.24)

Let Ω0(A, T ) be the set of these elements and let us say that G is the potential of F ∈ Ω0(A, T ).

Examples 1.7 i) If F ∈ Ω0(A, T ) then ω := dF ∈ Ω1(A, T ) since the both hand sides of (1.9)are zero. This Kahler form is closed and hence we have d : Ω0(A, T )→ Ω1(A, T ). In fact, (1.24)means that F ∈ Ω0(A, T ) if and only if the 1-form T (dF ) is exact.Moreover, due to the quadratic property of T it result the following duality: if F ∈ Ω0(A, T ) withthe potential G then G ∈ Ω0(A, T ) with the potential εF .ii) Suppose that F = R and A is the algebra C∞(M) of smooth functions on an even-dimensionalmanifold M2n; hence Ω1(A) is the usual C∞(M)-module of 1-forms on M and X 1(A) is the Liealgebra of vector fields on M . Let F be an almost complex or almost para-complex structure onM according to ε = −1 respectively ε = +1; recall that an almost para-complex structure is analmost product one with equal dimension for its±-eigenbundles. Then T is the dual endomorphismcorresponding to F :

Tω(X) := ω(FX) (1.25)

for all ω ∈ Ω1(M) and X ∈ X 1(M). Hence the definition 1 is exactly the definition 1.2 of [14, p.323] and thus we generalize the classical notion of almost analytic 1-form of [23]- [26].iii) Returning to the general case let ω1 and ω2 which are cohomological equivalent i.e. ω2 =ω1 + dF . Since dω2 = dω1 and d(Tω2) = d(Tω1) + d(T (dF )) it results that for ω1 ∈ Ω1(A, T ) we

5

have ω2 ∈ Ω1(A, T ) if and only if d T d(F ) = 0 . In particular, if F ∈ Ω0(A, T ) it results thatω1 and ω2 are simultaneous in Ω1(A, T ) or not. 2

In the following we develop an analogue of the Frolicher-Nijenhuis calculus. The first step isthe T -interior product iT : Ω•(A) → Ω•(A) given on 0-forms, 1-forms and exact 2-forms by therules:i) iTF = 0, iTω := Tω respectively,ii) Y (iX(iT dω)) := Y (iXdω) + Y (iXdω).The second step is the T -differential dT : Ω•(A)→ Ω•+1(A):

dT = iT d− εd iT . (1.26)

The i) above yields:iT iT = ε1Ω1(A) (1.27)

i.e. iT |Ω1(A) has the same nature as T , while (1.11) with ω → Tω gives the following expressionof ii):

Y (iX(iT dω)) = εY (iXd(Tω)) + Y (iXd(Tω)). (1.28)

These tools yield a stronger version of ii) of Proposition 1.2:

Proposition 1.8 I) For ω ∈ Ω1(A, T ) the following sentences are equivalent:i) ω is closed, ii) Tω is closed, iii) ω is T -closed: dTω = 0, iv) Tω is T -closed.II) If ω ∈ Ω1(A, T ) then ω is almost dT -analytic i.e. (1.9) holds also with d replaced with dT :

iXdTω = εiXdT (Tω). (1.29)

Proof I) Fix X,Y ∈ X 1(A). We have:

iXdTω = iX(iT dω)− εiXd(Tω) (1.30)

and then, with (1.10):

Y (iXdTω) = Y (iXdω) + Y (iXdω)− Y (iXdω) = Y (iXdω). (1.31)

It results the equivalence of i) and iii) and the conclusion follows.II) The relation (1.29) means:

iX(iT dω)− εiX(d(iTω)) = εıX [iT d(Tω)− εd(iTTω)].

equivalently:iX(iT dω)− εiXd(Tω) = εiX(iT d(Tω))− εiXdω.

This last equation holds term with term. 2

A simplification in the definition of iT on 2-forms is inspired by the Hermitian geometry(M2n, g, J) where the fundamental 2-form Ω := g(J ·, ·) satisfies:

Ω(JX, Y ) = −Ω(X, JY ) (1.32)

for any vector fields X,Y on the manifold M . Hence, for our setting we introduce:

6

Definition 1.9 The form ω ∈ Ω1(A) is called T -hermitian if:

Y (iXdω) = εY (iXdω) (1.33)

for any X,Y ∈ X 1(A). Let Ω1(A, Th) the F-module of these 1-forms.

Remarks 1.10 i) If ω ∈ Ω1(A, Th)∩Ω1(A, T ) then, after a straightforward calculus, we derive:

iX(iT dω) = (1 + ε)iXdω (1.34)

which for the complex case means: iT dω = 0 and dTω = d(Tω).ii) Let ω ∈ Ω1(A, T ). Then ω ∈ Ω1(A, Th) if and only if Tω ∈ Ω1(A, Th). Indeed, if ω ∈ Ω1(A, Th)then (1.33) with the terms replaced from (1.9) + (1.10) means:

εY (iXd(Tω)) = εY (iXd(Tω))

and hence:Y (iXd(Tω)) = Y (iXd(Tω)). (1.35)

With X → X we obtain:Y (iXd(Tω)) = εY (iXd(Tω)) (1.36)

which yields the claimed sentence. 2

In the following we study the expression of d2T for the diagram:

Ω0(A)dT−−→ Ω1(A)

dT−−→ Ω2(A) (1.37)

which means:Ω0(A) 3 F → dTF := ωF → dTωF := ρF . (1.38)

We have:ωT = iT (dF ) = T (dF ), ρF = iT (dωF )− εd(iTωF ) = iT (ddTF ). (1.39)

For F ∈ Ω0(A, T ) we obtain: d2TF = 0 while for the general case of F is necessary to consider the

2-form:µF = ddTF (1.40)

which yields:d2TF = iTµF . (1.41)

Inspired by the ddc-lemma of complex geometry, see for example [1, p. 75], we introduce:

Definition 1.11 The pair (A, T ) satisfies the ddT -Lemma if:

kerd ∩ imdT = imddT . (1.42)

Hence, if (A, T ) satisfies the ddT -Lemma there exists a closed 1-form ηF such that:

µF = dT ηF (1.43)

and we get the final expression of d2T :

d2TF = iT dT ηF = −εiT d(TηF ). (1.44)

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2 Applications to Riemann-Finsler geometry

Let N be now a smooth n-dimensional manifold with n ≥ 2 and π : TN → N its tangent bundle.Let x = (xi) = (x1, ..., xn) be local coordinates on N and (x, y) = (xi, yi) = (x1, ..., xn, y1, ...., yn)be the induced local coordinates on TN . Denote by O the null-section of π. A main structure ofTM is the tangent structure J = ∂

∂yi ⊗ dxi which satisfies also a quadratic equation: J2 = 0.

Recall after [3] or [5] that a Finsler fundamental function on N is a map F : TN → R+ withthe following properties:F1) F is smooth on the slit tangent bundle T0N := TN \O and continuous on O,F2) F is positive homogeneous of degree 1: F (x, λy) = λF (x, y) for every λ > 0,

F3) the matrix (gij) =(

12∂2F 2

∂yi∂yj

)is invertible and its associated quadratic form is positive definite.

The tensor field g = gij(x, y); 1 ≤ i, j ≤ n is called the Finsler metric and the homogeneity ofF implies:

F 2(x, y) = gijyiyj = yiy

i, (2.1)

where yi = gijyj . The pair (N,F ) is called Finsler manifold. In particular, if g does not depend

on y, we recover the Riemannian geometry.

On M := T0N we have two distributions:i) V (T0N) := kerπ∗, called the vertical distribution and not depending of F ; let v be the associatedprojector. It is integrable and has the basis ∂

∂yi ; 1 ≤ i ≤ n. A remarkable section of it is the

Liouville vector field C = yi ∂∂yi .

ii) H(T0N) with the basis δδxi := ∂

∂xi −Nji∂∂yj , where:

N ij =

1

2

∂γi00

∂yj(2.2)

with γi00 = γijkyjyk built from the usual Christoffel symbols:

γijk =1

2gia(∂gak∂xj

+∂gja∂xk

− ∂gjk∂xa

). (2.3)

A simple notation for γi00 is Gi. H(T0N) is often called the Cartan (or canonical) nonlinearconnection of the geometry (N,F ) and a remarkable section of it is the geodesic spray:

SF = yiδ

δxi. (2.4)

Let h be the associated projector. The dual basis of the above local basis δδxi ,

∂∂yi of Γ(T0N) is

(dxi, δyi = dyi +N ijdx

j). On T0N we have a Riemannian metric of Sasaki type:

GF = gijdxi ⊗ dxj + gijδy

i ⊗ δyj . (2.5)

Another Finslerian objects are two tensor fields of (1, 1)-type:I) [15] PF : Γ(T0N)→ Γ(T0N):

PF

(δ

δxi

)=

δ

δxi, PF

(∂

∂yi

)= − ∂

∂yi. (2.6)

8

Let us remark that PF is a global geometric object although it is defined using a fixed coordinatechart; in fact is exactly h− v and with equations (2.7) of [8, p. 14] we have:

PF J = −J, J PF = J. (2.7)

It results that PF is an almost para-complex structure and the pair (PF , GF ) is an almost para-Kahler structure on T0N .II) [16] ΨF : Γ(T0N)→ Γ(T0N):

ΨF

(δ

δxi

)=

∂

∂yi, ΨF

(∂

∂yi

)= − δ

δxi. (2.8)

It results that ΨF is an almost complex structure and the pair (ΨF , GF ) is an almost Kahlerstructure on T0N .

We have the following brackets:[δ

δxi,∂

∂yj

]=∂Nk

i

∂yj∂

∂yk,

[δ

δxj,δ

δxk

]= Rijk

∂

∂yi(2.9)

where the curvature R is:

Rijk =δN i

j

δxk− δN i

k

δxj. (2.10)

The geometry (N,F ) is called R-flat if R... = 0 which means the integrability of the distributionH(T0N); in the Riemannian case this means that the Riemannian metric g is flat. The tensorsPF , ΨF are not homogeneous and their homogeneous variant is studied in [6]. It is well-knownthat these ε-structures are integrable if and only if (N,F ) is R-flat. From (2.9) it follows alsothat: J [·, ·] = 0 on T0N .

The main 1-form of the Riemann-Finsler geometry is the Cartan 1-form [4, p. 410]:

ωF = dJF2 (2.11)

which is not closed; its differential dωF := θF is the symplectic form which yields the Hamiltoniannature of the geodesic spray with respect to the regular Lagrangian F 2. The 1-form: ω∗F := ωF ΨF

was considered in [11] under the name of ΨF -adjoint of ωF . The almost analyticity and thehermitian properties of ωF are given by:

Proposition 2.1 i) ωF is not almost analytic with respect to PF but is a PF -hermitian form.ii) ωF is not almost analytic with respect to ΨF but is a ΨF -hermitian form.

Proof i) ωF is almost analytic with respect to PF if and only if:

θF (PFX,Y ) = d(ωF PF )(X,Y ) (2.12)

for all X,Y ∈ Γ(T0N) which means:

PFX(JY (F 2))− J([PFX,Y ])(F 2) = X(J PFY (F 2))− J PF ([X,Y ])(F 2) (2.13)

or equivalently, with (2.7):

J([vX, Y ])(F 2)(= 0) = vX(JY (F 2)). (2.14)

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The only non-null case of vX and JY is provided by: X = ∂∂yj , Y = δ

δxi for which (2.14) means0 = 2gij impossible. The PF -hermitian property means:

PFX(JY (F 2))− Y (J PFX(F 2))− J([PFX,Y ])(F 2) +X(J PFY (F 2))− PFY (JX(F 2))−

−J([X,PFY ])(F 2) = 0 (2.15)

for all X,Y ∈ Γ(T0N); or equivalently, after a straightforward computation as before:

2hX(JY (F 2))− 2hY (JX(F 2)) = J([PFX,Y ] + [X,PFY ])(F 2) = 0. (2.16)

The only non-null case of hX, JX, hY, JY is provided by: X = δδxi , Y = δ

δxj for which (2.16)reads:

δ

δxj

(∂F 2

∂yi

)=

δ

δxi

(∂F 2

∂yj

)(2.17)

which, written again globally, means dhωF = 0. Equivalently, if X = hX and Y = hY then (2.16)means:

hX(JhY (F 2)) = hY (JhX(F 2)). (2.18)

ii) ωF is almost analytic with respect to ΨF if and only if:

θF (ΨFX,Y ) = −d(ω ΨF )(X,Y ) (2.19)

for all X,Y ∈ Γ(T0N); equivalently:

ΨFX(JY (F 2))− J([ΨFX,Y ])(F 2) = X(J ΨFY (F 2))− J ΨF ([X,Y ])(F 2). (2.20)

Since:J ΨF = −v (2.21)

the last equation is:ΨFX(JY (F 2)) = v([X,Y ])(F 2)−X(vY (F 2)). (2.22)

Due to the presence of both JY and vY we must discuss the both cases I: Y ∈ H(T0N) and II:Y ∈ V (T0N).I) let Y = δ

δxj . Then (2.22) is:

ΨFX

(∂F 2

∂yj

)= v

([X,

δ

δxj]

)(F 2) = [X,

δ

δxj](F 2) (2.23)

With X = δδxi the equation (2.23) reads: 2gij = Rkij

∂F 2

∂yk= Rkijyk, which is impossible since the

left hand side is 0-homogeneous while the right hand side is 2-homogeneous.II) let Y = ∂

∂yj . Hence (2.22) is:

0 = v

([X,

∂

∂yj]

)(F 2)−X

(∂F 2

∂yj

)(2.24)

which means:

X

(∂F 2

∂yj

)= [X,

∂

∂yj](F 2). (2.25)

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With X = ∂∂yi the equation (2.25) reads: 2gij = 0 again impossible.

The ΨF -hermitian property means:

ΨFX(JY (F 2))− Y (J ΨFX(F 2))− J([ΨFX,Y ])(F 2) +X(J ΨFY (F 2))−ΨFY (JX(F 2))−

−J([X,ΨFY ])(F 2) = 0 (2.26)

for all X,Y ∈ Γ(T0N) or equivalently:

ΨFX(JY (F 2)) + Y (vX(F 2)) = ΨFY (JX(F 2)) +X(vY (F 2)). (2.27)

I) let Y = δδxj . Then (2.27) is:

ΨFX(∂F 2

∂yj) +

δ

δxj(vX(F 2)) =

∂

∂yj(JX(F 2)). (2.28)

For X = δδxi we have the true equality 2gij = 2gij while for X = ∂

∂yi we have the true equality

(2.17).II) let Y = ∂

∂yj . Hence (2.27) is:

∂

∂yj(vX(F 2)) = − δ

δxj(JX(F 2)) +X

(∂F 2

∂yj

). (2.29)

For X = δδxi we have the true equality (2.17) while for X = ∂

∂yi we have the true equality2gij = 2gij . 2

The result of Proposition 2.1 means that from the point of view of both analyticity and thehermitian properties the endomorphisms PF and ΨF are similar although, as the computationsreveal, the para-complex structure is more easy to handle.

3 Almost para-CR and almost CR equations in a Riemann-Finsler geometry

Now, we study the almost analytic functions in Riemann-Finsler geometry:

Proposition 3.1 Let (N,F ) be a Riemann-Finsler geometry such that the 1-de Rham coho-mology space of tangent bundle is zero, H1(TN) = 0. Then:i) f ∈ C∞(T0N) is almost analytic with respect to PF if and only if the 1-form dPF f is closed,ii) f ∈ C∞(T0N) is almost analytic with respect to ΨF if and only if the 1-form dΨF f is closed.

Proof Due to the hypothesis regarding cohomology we have that the almost para-CR equationdf = df PF is equivalent with d(dPF f) = 0 while the almost CR-equation df = df ΨF isequivalent with d(dΨF f) = 0. Both differentials dPF , dΨF work on scalar fields from T0N . 2

Let us remark that in local coordinates:i) the almost para-CR equation df = df PF means:

∂f

∂yj= − ∂f

∂yj,

δf

δxi=

δf

δxi. (3.1)

11

ii) the almost CR-equation df = df ΨF means:

∂f

∂yj= − δf

δxj,

δf

δxi=

∂f

∂yi. (3.2)

and inspired by Proposition 3.1 we introduce a weaker variant of almost analyticity through:

Definition 3.2 For an arbitrary Riemann-Finsler geometry (N,F ) and f ∈ C∞(T0N):i) f is called weak almost PF -analytic function if d(dPF f) = 0, ii) f is called weak almost ΨF -analytic function if d(dΨF f) = 0.

We have: d(dPF f)( ∂

∂yi ,∂∂yj ) = 0, d(dPF f)( δ

δxi ,δδxj ) = 2Raij

∂f∂ya ,

d(dPF f)( ∂∂yj ,

δδxi ) = 2 ∂

∂yj

(δfδxi

) (3.3)

d(dΨF f)( ∂

∂yi ,∂∂yj ) = − ∂

∂yi

(δfδxj

)+ ∂

∂yj

(δfδxi

),

d(dΨF f)( δδxi ,

δδxj ) = δ

δxi

(∂f∂yj

)− δ

δxj

(∂f∂yi

)+Raij

δfδxa ,

d(dΨF f)( ∂∂yj ,

δδxi ) = δ2f

δxjδxi + ∂2f∂yj∂yi −

∂2Ga

∂yj∂yiδfδxa

(3.4)

and hence we derive:

Proposition 3.3 i) The function f ∈ C∞(T0N) is weak almost PF -analytic function if andonly if the functions δf

δxi depends only on x i.e. are functions on the base N and:

Raij∂f

∂ya= 0. (3.5)

In particular, if H1(T0N) = 0 then: i1) any f ∈ C∞(N) is almost analytic with respect to PFwith fC = f +C ∈ C∞(N) for an arbitrary constant C; hence the 1-form df is an +1-eigenvectorfor P ∗F .i2) Moreover, if (N,F ) is R-flat then f = ϕ(F ) with an arbitrary smooth ϕ : R+ → R is analyticwith respect to PF with:

f(x, y) = −∫ 1

t=0

ϕ′(F (x, ty))F (x, ty)dt. (3.6)

ii) The function f ∈ C∞(T0N) is weak almost ΨF -analytic function if and only if:∂∂yi

(δfδxj

)= ∂

∂yj

(δfδxi

),

δδxi

(∂f∂yj

)− δ

δxj

(∂f∂yi

)+Raij

δfδxa = 0,

δ2fδxjδxi + ∂2f

∂yj∂yi −∂2Ga

∂yj∂yiδfδxa = 0.

(3.7)

Example 3.4 (Riemannian geometry) Suppose that (N, g) is a Riemannian manifold. Then(3.5) is:

Raijk(x)yk∂f

∂ya(x, y) = 0 (3.8)

with R.... the Riemannian curvature tensor field of g. Also, (3.73) becomes:

δ2f

δxjδxi+

∂2f

∂yj∂yi= γaji(x)

δf

δxa(3.9)

12

and if we search for functions f on the base N we obtain:

∂2f

∂xi∂xj= γaij

∂f

∂xa(3.10)

which means that the Hessian of f with respect to g vanishes i.e. f are the totally geodesicfunctions of the geometry (N, g). It results immediately the expression of f from (3.2):

f(x, y) = − ∂f∂xi

(x)yi = −fC(x, y) (3.11)

where fC means the complete lift of f to TN . The condition (3.8) or (3.72) means for a basefunction f = f(x):

Raijk(x)∂f

∂xa(x) = 0 (3.12)

and we remark that in the flat case this condition is satisfied.

We can provide an interpretation for (3.12) as follows: for any vector fields X,Y, Z on N theapplication of the Riemannian curvature R(·, ·)· yields the vector field R(X,Y )Z; hence (3.12)means that f is a first integral for all the vector fields R(X,Y )Z. In conclusion, if for any x ∈ Nthe curvature Rx(·, ·)· spans the whole tangent space TxN then h must be a constant; this is thecase of non-flat space-forms. The same formula (3.12) is globally expressed as dh R = 0 and itis obtained in [7, p. 175] as the characterization condition which assures that the vertical lift of atotally geodesic function h is totally geodesic with respect to GF . 2

Inspired by this example we introduce two new types of Hessian for f ∈ C∞(T0N) in aRiemann-Finsler geometry:a) the almost para-complex Hessian:

PCHfij := 2

∂

∂yi

(δf

δxj

)= 2d(dhf)

(∂

∂yi,δ

δxj

)(3.13)

with dh the horizontal component of the total differential d of T0N i.e. the differential with respectto the horizontal projector h.b) the almost complex Hessian:

CHfij :=

δ2f

δxjδxi+

∂2f

∂yi∂yj− ∂2Ga

∂yi∂yjδf

δxa. (3.14)

Remarks 3.5 i) Let us point out that another expression for these operators is:

PCHij =∂

∂yi PF

∂

∂xj− ∂

∂xj PF

∂

∂yi, CHij =

∂

∂yiΨF

∂

∂xj− ∂

∂xjΨF

∂

∂yi(3.15)

and an unpleasant characteristic of them is the non-symmetry; the Hessian PCHf is symmetricif and only if (3.71) holds while the Hessian CH is symmetric if and only if the geometry (N,F )is R-flat.

Let us point out that (3.71) holds if f is the complete lift hC of a smooth base functionh ∈ C∞(N) since in this case we have:

∂

∂yi

(δf

δxj

)=

∂2h

∂xi∂xj− ∂2Ga

∂yi∂yj∂h

∂xa=

∂

∂yj

(δf

δxi

)(3.16)

13

and again, in the Riemannian case, the middle term is the Hessian of h.ii) The contraction of these Hessians with yi and yj yields the following scalar fields:

PCHf00 := 2yiyjPCHf

ij = 2SF (C(f)), CHf00 := yiyjCHf

ij = SF (SF (f)) +C(C(f))−C(f) (3.17)

where we use the 2-homogeneity of SF , [C, SF ] = SF , and that of Ga: Nauy

u = 2Ga. In particular,if f is r-homogeneous with respect to y i.e. C(f) = rf then:

PCHf00 = 2rSF (f), CHf

00 = SF (SF (f)) + (r2 − r)f (3.18)

and if f is a Rayleigh dissipation function of SF , namely SF (f) < 0 according to [9, p. 1558],

then PCHf00 has the opposite sign of r.

iii) Concerning with the second part of (3.13) let us remark that for any f ∈ C∞(T0N):

d(dPF f) = 2d(dhf) (3.19)

and hence we obtain another characterization: f ∈ C∞(T0N) is weak almost PF -analytic if andonly if d(dhf) = 0.

It follows that if f is horizontally constant i.e. dhf = 0 then f is weak almost PF -analytic. Forexample F is horizontally constant and if the geodesic spray SF is Ricci-constant conform [4, p.407] then the Ricci scalar function R ∈ C∞(T0N), defined in the cited paper as the trace of theJacobi tensor (see below), is also horizontally constant. 2

It follows also new types of Laplacians as trace of these Hessians:c) the almost para-complex Laplacian:

PC∆(f) := TrgPCHf = gijPCHf

ij = 2gij∂

∂yi

(δf

δxj

)(3.20)

d) the almost complex Laplacian:

C∆(f) := TrgCHf = gijCHf

ij = gij(

δ2f

δxjδxi+

∂2f

∂yi∂yj− ∂2Ga

∂yi∂yjδf

δxa

). (3.21)

In [17, p. 68] is affirmed that this is exactly the Riemannian Laplacian with respect to theRiemannian metric GF . We discuss this statement in the following section.

Examples 3.6 i) We have PC∆(F 2) = 0 and C∆(F 2) = 2n since CHF 2

ij = 2gij ; so F 2 is analmost para-complex-harmonic function and we ”hear” the dimension on T0N . These equalities aresimilar to the Proposition 3.3. of [27, p. 134] where for a horizontal respectively vertical Laplacianon (N,F ) is obtained ∆hF

2 = 0 and ∆vF2 = 2n. Moreover, every horizontally constant function

is almost para-complex-harmonic.ii) (Euclidean geometry) Let (N, g) = (Rn, can) be the n-dimensional Euclidean space and f ∈C∞(TRn) = C∞(R2n). Then:

PC∆(f) = 2

n∑i=1

∂2f

∂xi∂yi, C∆(f) =

n∑i=1

(∂2f

∂(xi)2+

∂2f

∂(yi)2

)= ∆(R2n,can)f (3.22)

and hence:ii1) the para-complex-harmonic functions on T (Rn, can) = (R2n, can) are: f(x, y) = a(x) + b(y)

14

with smooth a and b,ii2) the complex-harmonic functions on T (Rn, can) = (R2n, can) are exactly the Euclidean har-monic functions.Let < ·, · >n be the Euclidean inner product on the base Rn and two fixed vectors α, β ∈ Rn.Then the function:

fα,β(x, y) := exp(< α, x >n + < β, y >n)

is an eigenfunction of PC∆ with the eigenvalue < α, β >n and for C∆ with the eigenvalue‖α‖2n + ‖β‖2n.iii) (Berwald and Landsberg geometries) Recall after [5, p. 38] that a Finsler geometry is a Berwaldone if Gi = Γiuv(x)yuyv and this is the more closed to Riemannian geometry from several reasonsone of them being that Riemannian metrics are Berwald; but there exist non-Riemannian Berwaldmanifolds. Then in such a geometry we have: PC∆(f) = 2gij

[∂2f

∂yi∂xj − Γaij(x) ∂f∂ya − Γaju(x)yu ∂2f∂ya∂yi

],

C∆(f) = gij[

δ2fδxjδxi + ∂2f

∂yi∂yj − Γaij(x) δfδxa

].

(3.23)

On the other hand let us denote, following [19], with ∇HC the horizontal covariant derivative forthe Cartan or Chern-Rund connection of (N,F ). Then on page 6 of the cited preprint is computedthe divergence of a vertical gradient Z = (Zi = ∂f

∂yi ) := gradvf as follows:

∇HC · Z = gij[∂

∂yj

(δf

δxi

)+ Laij

∂f

∂ya

](3.24)

where L is the Landsberg tensor field of (N,F ). The geometry (N,F ) is a Landsberg one if L = 0and by Proposition 2.1.3 of [5, p. 39] every Berwald geometry is Landsberg. In conclusion, in aLandsberg, particularly Berwald (more particularly Riemann), geometry we have a global formulafor PC∆:

PC∆(f) = ∇HC · gradv(2f). (3.25)

From this formula we can introduce the weighted case of PC∆ following [12]. Let µ ∈ C∞(T0N)with µ > 0. The weighted divergence of Z is:

∇HCµ · Z :=1

µ∇HC · (µZ)

and hence the weighted almost para-complex Laplacian is:

PC∆µ(f) := ∇HCµ · gradv(2f) =1

µ∇HC · µgradv(2f).

In particular, if µ = µ(x) then:

PC∆µ(f) =1

µ∇HC · gradv(2µf) =

1

µPC∆(µf) = PC∆(f) +

2

µgij

∂µ

∂xj∂f

∂yi.

For example: PC∆µ(F 2) = 4µµ

C .

iv) (Eigenvalue problems) Let us search f = ϕ(F 2) as eigenfunction for C∆ with a smoothϕ : R∗+ → R. Since:

C∆(f) = 4ϕ′′F 2 + 2nϕ′ (3.26)

15

it follows that ϕ = ϕ(t) solves the eigenvalue problem:

4tϕ′′ + 2nϕ′ = λϕ (3.27)

for λ ∈ R. Multiplying with tn2−1 we arrive at:

(tn2 ϕ′)′ =

λ

4tn2−1ϕ (3.28)

and for strictly negative λ’s we have the general solution ϕ(t) = C1J0(√−λt)+C2Y0(

√−λt) where

C1, C2 are constants and J0, Y0 are the Bessel functions of the first and second kind respectively.For the eigenvalue λ = 0 it follows the eigenfunction:

ϕn≥3(t) =2C

2− nt1−

n2 , ϕn=2(t) = C ln t, (3.29)

with C an arbitrary constant. In conclusion, the function f = 2C2−nF

2−n, respectively f = 2C lnF ,is both almost para-complex-harmonic and almost complex-harmonic. Let us remark that λ = 0in (3.27) together with n = 1 means 2tϕ′′+ϕ′ = 0 which, after [2, p. 216], means that L = ϕ(F 2)is a singular Lagrangian on TN ; this is a reason for our choice n ≥ 2 of Section 2.v) Closely related to the harmonicity is the notion of Dirichlet energy density: in a Riemanniangeometry (N, g) the energy density of h ∈ C∞(N) is:

e(f) :=1

2|df |g =

1

2gij

∂f

∂xi∂f

∂xj(3.30)

and the harmonic functions are the critical points of this functional. Hence in a Riemann-Finslergeometry (N,F ) for f ∈ C∞(T0N) its energy density is:

eF (f) :=1

2|df |GF =

1

2gij(δf

δxiδf

δxj+∂f

∂yi∂f

∂yj

). (3.31)

For example eF (F 2) = 2F 2. Suppose now that (N,F = Fg) is the Riemannian geometry of aRiemannian metric g on N and f = hC . A straightforward computation yields:

eFg (hC) =1

2gij(∂h

∂xi∂h

∂xj+Hh

iuHhjvy

uyv)

(3.32)

with Hh the Hessian of h with respect to g. One obtains that if h is totally geodesic with respectto g then:

eFg (hC) = eg(h). (3.33)

Returning to the general case of (N,F ) we point that e(f) arises naturally from the relation:

C∆(f2) = 2fC∆(f) + 4e(f). (3.34)

2

We introduce now two types of gradient Ricci solitons in the tangent bundle of a Riemann-Finsler geometry. There are necessary some notations. More precisely, the Jacobi endomorphismof (N,F ) is defined in [4, p. 407] as:

Φ = v LSF h = LSF h h (3.35)

16

with local expression:

Φ = Rij∂

∂yi⊗ dxj =

[2∂Gi

∂xj− SF

(∂Gi

∂yj

)− ∂Gi

∂ya∂Ga

∂yj

]∂

∂yi⊗ dxj (3.36)

and then the Ricci scalar function is: Ric = Tr(Φ) = Rii. Finally, the Ricci tensor field of (N,F )is:

Ricij =1

2

∂2(F 2Ric)

∂yi∂yj(3.37)

and hence we introduce:

Definition 3.7 1) The symmetric almost para-complex Hessian is:

PCsHij :=1

2(PCHij + PCHji) =

∂

∂yi

(δ

δxj

)+

∂

∂yj

(δ

δxi

). (3.38)

The symmetric almost complex Hessian is:

CsHij :=1

2(CHij + CHji) =

=1

2

[δ

δxi

(δ

δxj

)+

δ

δxj

(δ

δxi

)]+

∂2

∂yi∂yj− ∂2Gk

∂yi∂yjδ

δxk. (3.39)

2) Let f ∈ C∞(T0N) and λ ∈ R. Then:i) the pair (f, λ) is a weak almost para-complex gradient Ricci soliton for (N,F ) if:

PCsHfij +Ricij + λgij = 0, (3.40)

ii) the pair (f, λ) is a weak almost complex gradient Ricci soliton for (N,F ) if:

CsHfij +Ricij + λgij = 0. (3.41)

A contraction with yi, yj yields, via (3.17):

2SF (C(f)) + F 2(Ric+ λn) = 0, SF (SF (f))− C(C(f))− C(f) + F 2(Ric+ λn) = 0 (3.42)

and we finish this section with a class of such Ricci solitons provided by the constancy of the flagcurvature:

Proposition 3.8 Let (N,F ) be a Riemann-Finsler geometry of constant flag curvature c andsuppose that f is a weak almost para-complex (respectively weak almost-complex) totally geodesicfunction. Then the pair

(f, λ = −n−1

n c)

is a weak almost para-complex (respectively complex)

gradient Ricci soliton. In particular,(

22−nF

2−n, 1−nn c)

for n ≥ 3 or (2 lnF,− c2 ) for n = 2 is both

weak almost para-complex and weak almost complex gradient Ricci soliton.

Proof If (N,F ) has the constant flag curvature then:

Ric = (n− 1)c, Ricij = (n− 1)cgij (3.43)

and the conclusion follows directly. The last part is a consequence of iv) of Examples 3.6. 2

17

4 Hessian and Laplacian on the tangent bundle of a Riemann-Finsler geometry

Let us denote by F∇ the Levi-Civita connection of the Riemannian metric GF and by FHf thecorresponding Hessian of f ∈ C∞(T0N). Hence:

FHf (X,Y ) = X(Y (f))− F∇XY (f) (4.1)

for every vector fields X,Y on T0N and in particular we have the horizontal and vertical compo-nents:

hFHfij := FHf

(δ

δxj,δ

δxi

), vFHf

ij := FHf

(∂

∂yj,∂

∂yi

). (4.2)

It follows:

hFHfji =

δ2f

δxjδxi− F∇ δ

δxj

δ

δxi(f), vFHf

ji =∂2f

∂yjδyi− F∇ ∂

∂yj

∂

∂yi(f) (4.3)

and hence the Riemann-Finsler Hessian of f is:

F∆(f) = gij(hFHfji + vFHf

ji) = gij[

δ2f

δxjδxi+

∂2f

∂yi∂yj− F∇ δ

δxj

δ

δxi(f)− F∇ ∂

∂yj

∂

∂yi(f)

](4.4)

and the expression of F∇ appears in [3, p. 228]. Comparing with our (3.21) it results a verydifferent expression; for example F∆(F 2) = 2n+ gijRaijgaky

k.

But we can express our Hessian CHf using the theory of Finslerian connections which aretriples Γ = (Nk

i , Fkij(x, y), Ckij(x, y)) where F ... behave like the coefficients of a linear connection

and C is a tensor field on T0N . Such a Finslerian connection yields the linear connection Γ∆ onT0N given by:

Γ∆ δ

δxj

δδxi := F kij

δδxk

, Γ∆ δ

δxj

∂∂yi := F kij

∂∂yk

Γ∆ ∂

∂yj

δδxi := Ckij

δδxk

, Γ∆ ∂

∂yj

∂∂yi := Ckij

∂∂yk

.(4.5)

It follows the corresponding Hessian again with horizontal-horizontal and vertical-vertical compo-nents:

hΓHfji := ΓHf

(δδxj ,

δδxi

)= δ2f

δxjδxi − Fkijδfδxk

,

vΓHfji := ΓHf

(∂∂yj ,

∂∂yi

)= ∂2f

∂yj∂yi − Ckij∂f∂yk

(4.6)

which is symmetric if and only if (N,F ) is flat and Γ∆ is h- and v-symmetrical i.e. F kij = F kjirespectively Ckij = Ckji. The vertical-horizontal component of the Hessian is:

vhΓHfij := ΓHf

(∂

∂yi,δ

δxj

)=

∂

∂yi

(δf

δxj

)− Ckij

δf

δxk. (4.7)

We derive also the corresponding Laplacians:

Γ∆(f) = gij(hΓHf

ji + vΓHfji

), Γ∆vh(f) = gijvhΓHf

ij . (4.8)

There are four remarkable Finslerian connections, [3, p. 227]:-Cartan Ca = (Nk

i , Fkij , C

kij),

18

-Chern-Rund CR = (Nki , F

kij , 0),

-Berwald B = (Nki , G

kij , 0),

-Hashiguchi H = (Nki , G

kij , C

kij),

where:

F kij =1

2gka

(δgajδxi

+δgiaδxj− δgijδxa

), Ckij =

1

2gka

∂gij∂ya

, Gkij =∂2Gk

∂yi∂yj. (4.9)

In conclusion we have six remarkable Laplacians in addition to the Riemann-Finsler LaplacianF∆:

Ca∆(f) = gij(

δ2fδxjδxi + ∂2f

∂yj∂yi − Fkijδfδxk− 1

2gka ∂gij

∂ya∂f∂yk

)CR∆(f) = gij

(δ2f

δxjδxi + ∂2f∂yj∂yi − F

kijδfδxk

)B∆(f) = gij

(δ2f

δxjδxi + ∂2f∂yj∂yi −

∂Gk

∂yi∂yjδfδxk

)H∆(f) = gij

(δ2f

δxjδxi + ∂2f∂yj∂yi −

∂Gk

∂yi∂yjδfδxk− 1

2gka ∂gij

∂ya∂f∂yk

)Ca∆vh(f) = H∆vh(f) = gij

[∂∂yi

(δfδxj

)− Ckij

δfδxk

]CR∆vh(f) = B∆vh(f) = gij ∂

∂yi

(δfδxj

).

(4.10)

It results that our almost para-complex Laplacian is 2CR∆vh = 2B∆vh and the almost complexLaplacian C∆ is exactly the Berwald Laplacian B∆.

These Laplacians have the same value on F 2:PC∆(F 2) = Ca∆vh(F 2) = CR∆vh = B∆vh(F 2) = H∆vh(F 2) = 0,Ca∆(F 2) = CR∆(F 2) = B∆(F 2) = H∆(F 2) = 2n

(4.11)

and since the characterization of Landsberg geometry is F kij = Gkij , conform [3, p. 230], it followsthat:

Landsberg : Ca∆ = H∆, CR∆ = B∆. (4.12)

In the particular case of Riemannian geometry we have F kij = Gkij = γkij(x) and then:

Riemannian : Ca∆(f) = CR∆(f) = B∆(f) = H∆(f) = gij(

δ2f

δxjδxi+

∂2f

∂yi∂yj− γaij

δf

δxa

).

(4.13)For example let f = hC and denotes Hh the Hessian of h ∈ C∞(N) with respect to the Riemannianmetric g(x). Then:

·∆(hC) = gijya(∂Hh

ia

∂xj− γkjaHh

ik − γkijHhka

)(4.14)

and we recognize in the right-hand-side the Christoffel process of the metric g. Hence if Hh isparallel with respect to g, in particular h is totally geodesic with respect to g or Hh is multiple ofg, then hC is harmonic with respect to ·∆.

Acknowledgement The authors are grateful to Ioan Bucataru for several discussions onFinsler geometry and a major simplification in the proof of Proposition 2.1. Also we express ourgratitude to two anonymous referees for several excellent remarks and improvements.

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Mircea Crasmareanu Laurian-Ioan PiscoranFaculty of Mathematics Department of Mathematics and Computer ScienceUniversity ”Al. I.Cuza” North University Center of Baia Mare, Technical University of ClujIasi, 700506 Baia Mare, 430122Romania RomaniaEmail: mcrasm@uaic.ro Email: plaurian@yahoo.comhttp://www.math.uaic.ro/∼mcrasm

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