Finslerian slit tangent bundle

Finslerian slit tangent bundle

G. Nibaruta, A. Nibirantiza, M. Karimumuryango and D. Ndayirukiye

Abstract.LetF be a Finslerian metric on aC^∞manifoldM. We define horizontal and vertical Laplace type operators forC^∞functions on the slit tangent bundle ˚T M of the Finslerian manifold (M, F). The expression of the vertical Laplacian ∆^v, in local coordinates, is obtained by using the Chern connection on the pulled-back tangent bundle. Furthermore, we prove the vertical divergence lemma and we show that if (M, F) is an oriented Finslerian manifold then every globally defined functionuon ˚T M, with ∆^vu≥0 everywhere or ∆^vu≤0 everywhere, must be independant of directional arguments.

M.S.C. 2010: 31A05, 53C60, 55R25.

Key words: Finslerian manifolds; pulled-back tangent bundle; Chern connection;

vertical Laplacian.

1 Introduction

The geometry of the vector bundles began in the 1950s with Sasaki, who built a metric on the tangent bundle of a Riemannian manifold [1]. Let consider a Finslerian manifold (M, F) and ˚T M ≡T M\{0} its slit tangent bundle. The Finslerian metric F induces on ˚T M a fundamental tensor g = gij(x, y)dxⁱ⊗dx^j, with (x, y) ∈ T M˚ whose (xⁱ, yⁱ) is its representation in local coordinate. As explain in [3], the tensor g defines at every point (x, y)∈T M˚ an inner product on each tangent spaceTxM of the manifoldM.

The Laplace operator plays an important role in the theory of harmonic maps.

This last subject has found many applications. More informations on harmonic maps can be found: in [7] for Euclidean spaces particularly for harmonic functions and in [2]

for Riemannian manifolds. Hence, questions about aspects of harmonic (subharmonic, superharmonic) functions for Finslerian manifolds arise. In Finslerian geometry, the Laplace type operator can be defined in different ways and is made either onM or on ˚T M. See [6] for Laplacians on ˚T M in terms of Cartan connection and [10, 11]

for Laplacians on M in terms of Chern connection. The Chern connection is, for

Balkan Journal of Geometry and Its Applications, Vol.25, No.1, 2020, pp. 93-103.∗

⃝c Balkan Society of Geometers, Geometry Balkan Press 2020.

us, the most important Finslerian connection. Note that, as introduced in the first paragraph of this Section, the componentsgij of g, associated with F, depend not only on the base pointx∈M, as in the Riemannian case, but also on the direction y∈TxM. Then, it is very important to define a Laplacian operator for functions on T M˚ of (M, F).

In this paper, we use the pulled-back tangent bundle approach [3, 12] to define vertical Laplacian operator, denoted by ∆^v, on ˚T M. Our main goal is to generalize the classical Riemannian E. Hopf’s theorem, given in [3] for harmonic (subharmonic, superharmonic) functions. With respect to the horizontal Laplacian, the analogous of this theorem refers to [6] where it is proved by applying Cartan ideas and by decomposition of the Laplace-Beltrami operator on (˚T M, g^s) whereg^sis the Sasakian metric.

The rest is organised as follows: in the second Section, we recall some basic notions on Finslerian manifolds which are used throught this paper. In Section 3, with the Chern connection on the pulled-back tangent bundleπ^∗T M by the submersion π : T M˚ −→M, we define ∆^v forC^∞-functions on ˚T M and give some of its properties.

In Section 4, we give a vertical divergence formula for sections ofπ^∗T M. Thus, we prove our main results.

2 Some basic notions

Throughout this paper,M is ann−dimensional connectedC^∞ manifold. We denote by TxM the tangent space at x ∈ M, by T M := ∪

x∈MTxM the tangent bundle ofM and by SM :={(x,[y])} the sphere bundle of M. Set ˚T M =T M\{0}. The natural projection π : T M −→ M is given by π(x, y) = x. Let (x¹, ..., xⁿ) be a local coordinate on an open subset U of M and (x¹, ..., xⁿ, y¹, ..., yⁿ) be the local coordinate on π⁻¹(U) ⊂ T M. The said (xⁱ), with i = 1, ..., n, produce the basis sections{_∂x^∂i} and{dxⁱ}, respectively, for T M and cotangent bundleT^∗M. We use Einstein summation convention.

2.1 Finslerian manifolds

Definition 2.1. AFinslerian metriconM is a functionF :T M −→[0,∞) with the following properties :

(1) F isC^∞ on the entire slit tangent bundle ˚T M.

(2) F is positively 1-homogeneous on the fibers ofT M, that is

∀c >0, F(x, cy) =cF(x, y).

(3) the Hessian matrix (gij(x, y))1≤i,j≤n with elements g_ij(x, y) :=∂²[₁

2F²(x, y)]

∂yⁱ∂y^j (2.1)

is positive-definite at every point (x, y) of ˚T M.

Given a manifoldM and a Finslerian metricF onT M, the pair (M, F) is called aFinslerian manifold.

2.2 The vector bundle π

T M and related objects

Let M be a connected C^∞ manifold. The restricted projection π : ˚T M −→ M pulles backT M to a bundleπ^∗T M over ˚T M called thepulled-back tangent bundle.

π^∗T M is a vector bundle over the slit tangent bundle ˚T M whose fiber (π^∗T M)|(x,y)

at (x, y)∈T M˚ is just a copy of the tangent spaceTxM. Then (π^∗T M)|(x,y):={(x, y, ξ) :ξ∈TxM} ∼=TxM.

(2.2) Set

g_ij(x, y) :=1 2

∂²F²(x, y)

∂yⁱ∂y^j (2.3)

and

Aijk(x, y) := F 2

∂gij(x, y)

∂y^k = F 4

∂³F²(x, y)

∂yⁱ∂y^j∂y^k. (2.4)

One obtains two tensors g = gijdxⁱ ⊗dx^j and A = Aijkdxⁱ ⊗dx^j ⊗dx^k called respectively fundamental tensor and Cartan tensor. g is a symmetric section of π^∗T^∗M ⊗π^∗T^∗M whileAis a symmetric section ofπ^∗T^∗M⊗π^∗T^∗M⊗π^∗T^∗M. Lemma 2.1. Let (M, F)be ann-dimensional Finslerian manifold and Athe Cartan tensor. Then, for alli, j, k∈ {1, ..., n} and(x, y)∈T M,˚

(i) Aijk=Aikj=Ajik.

(ii) yⁱAijk(x, y) =y^jAijk(x, y) =y^kAijk(x, y) = 0.

(iii) (M, F)is Riemannian if and only if Aijk(x, y) = 0.

Now, consider all points in ˚T M of the form (x, cy), with x, y fixed and c an arbitrary positive real number. Over each such point, one erects the same vector space T_xM. Because the componentsg_ij(x, y) ofgare invariant under the rescalingy7→cy then the inner products, assigned to the copies (π^∗T M)|(x,cy)ofT_xM, are also identic.

One setSM :={(x,[y]) : (x, y)∈T M˚ }, where [y] :={cy:c >0, y∈T_xM\{0}}, and call thesphere bundlewhich is the (2n−1)-dimensional manifold overM.

Remark 2.2. We work on ˚T M. But, in order to use objects that have the same sense following that one works on ˚T M orSM, we focus on objects which are invariant by transformationy 7−→cy, c >0. For example, we use Aikj and ^δy_Fⁱ instead of _F¹Aikj

and δyⁱ respectively. Since the objects are invariant by rescaling in y, they can be seen as carried out onSM using homogeneous coordinates.

2.3 Finslerian Ehresmann connection

Consider the tangent mapping π_∗ of the restricted projection π : ˚T M −→ M : π(x, y) 7−→ x. The vertical subspace of TT M˚ is defined by V := ker(π_∗) which is locally spanned by the set{F_∂y^∂_i,1≤i≤n}, on eachπ⁻¹(U)⊂T M˚ .

An horizontal subspaceHofTT M˚ is by definition any complementary toV. The bundlesHandV give a smooth splitting [9]

TT M˚ =H ⊕ V. (2.5)

An Ehresmann connection is a selection of horizontal subspaceHofTT M.˚

All Finslerian metric F on M induces a vector fields on ˚T M [4] in the form G=y^{i ∂}_∂xi−2G^{i ∂}_∂yi where the elements

Gⁱ(x, y) := 1 4g^il

(∂gjl

∂x^k +∂glk

∂x^j −∂gjk

∂x^l )

y^jy^k

arey-homogeneous of degree two. The vector field Gis called spray on M and the Gⁱ are called spray coefficientsofG. Consider the functions

N_jⁱ(x, y) :=∂Gⁱ(x, y)

∂y^j .

One says thatN_jⁱ(x, y) areEhresmann connection coefficientson manifold ˚T M. One has

N_jⁱ=−1

FAjklg^ilγ_rs^ky^ry^s+γ_jkⁱ y^k, i, j, k, r, s= 1, ..., n (2.6)

where γ_jkⁱ := ¹₂g^il (∂gjl

∂x^k +^∂g_∂x^lkj −^∂g_∂x^jkl

)

are formal Christoffel symbols of the second kind.

Remark 2.3. According to the remark 2.2, for objects invariant undery 7→cy, we can considerN_jⁱ as Ehresmann connection coefficients onSM. In that case, we must work with ^N

i j

F :=−Ajklg^ilγ^k_rs^y_F^ry_F^s +γ_jk^{i y}_F^k.

Consider the local coordinate (xⁱ, yⁱ) in T M. One has, respectively, the local coordinates bases {_∂x^∂i,_∂y^∂i} for the tangent bundle of T M and {dxⁱ, dyⁱ} for the cotangent bundle of T M. For the tangent bundle of ˚T M, a local coordinate basis adapted to the above decomposition is consists of the _δx^δ_i := _∂x^∂_i −N_i^j_∂y^∂_j and _∂y^∂_i. The dual of the two last basis elements are respectivelydxⁱ andδyⁱ :=dyⁱ+N_jⁱdx^j. In the sequel, we consider the Ehresmann connection (called Finslerian Ehresmann connection) defined as follows

Definition 2.4. A Finslerian Ehresmann connection of the restricted projectionπis a subspaceHofTT M˚ , which is complementary to the vertical subspaceV ofTT M˚ , given byH:=kerθ whereθ is, globally, a C^∞ function fromTT M˚ to π^∗T M called theEhresmann form. Locally, θis given by

θ= ∂

∂xⁱ ⊗ 1

F(dyⁱ+N_jⁱdx^j).

(2.7)

Proposition 2.2. [12] Given a Finslerian manifold (M, F) and g the fundamental tensor associated with Finslerian metric F, let π_∗ be the tangent mapping of the

restricted projectionπ: ˚T M −→M. There exist a unique linear connection∇ on the pulled-back tangent bundleπ^∗T M,

∇: Γ(TT M˚ )×Γ(π^∗T M) −→ Γ(π^∗T M) (X , ξ) 7−→ ∇Xξ

such that, for allX, Y ∈Γ(TT M)˚ andξ, η ∈Γ(π^∗T M), one has the following prop- erties:

(i) Symmetry: ∇Xπ_∗Y − ∇Yπ_∗X=π_∗[X, Y],

(ii) Almost g-compatibility: X(g(ξ, η)) =g(∇Xξ, η) +g(ξ,∇Xη) + 2A(θ(X), ξ, η), whereAis the Cartan tensor and θis the Ehresmann form given by (2.7).

2.4 Riemannian metric on the slit tangent bundle T M ˚

The slit tangent bundle ˚T M has a natural Riemannian metric g^s=g_ijdxⁱ⊗dx^j+g_ijδyⁱ

F ⊗δy^j F , (2.8)

known (see [3], Section 2.1) as a Sasaki (type) metric, whereg_ijare the components of the fundamental tensorg. With respect to this metric, we havespan{_δx^δi}⊥span{F_∂y^∂_i}, that is

TT M˚ = H ⊕ V

= span { δ

δxⁱ }

⊕span {

F ∂

∂yⁱ }

. (2.9)

Remark 2.5. The y-homogeneous of degree zero of g induces the invariance ofg^s under the positive rescaling ofyandg^scan be considered as a Riemannian metric on SM.

3 Fundamental differential operators on T M ˚

Let (M, F) be an n-dimensional Finslerian manifold, g be the fundamental tensor ofF and g^s be the Sasakian (type) metric on the slit tangent bundle ˚T M. In this Section, we define fundamental differential operators on ˚T M which are used in the following.

3.1 Gradient section of the vector bundle π

T M

For the Riemannian manifold (˚T M, g^s), the gradient of a functionu∈ C^∞(˚T M) is given by:

g^s(Ou, X) =X(u), ∀X∈Γ(TT M˚ ).

(3.1) Let

{ δ δxⁱ, F_∂y^∂i

}

be the corresponding basis to the direct sum of the horizontal H and the verticalV subspaces ofTT M˚ , whereF_∂y^∂_i is the homogenized usual partial

derivative. Using a local coordinate (xⁱ, yⁱ) in ˚T M, every sectionX ofTT M˚ is given byX = (X^h)^{i δ}_δxi + (X^v)ⁱF_∂y^∂i where the (X^h)ⁱ and the (X^v)ⁱ areC^∞ functions on T M. Then, locally, we have˚

Ou=g^ij δu δxⁱ

δ

δx^j +g^ijF²∂u

∂yⁱ

∂

∂y^j. (3.2)

It is known, the vertical space V, as well as the horizontal H, can be naturally identified withπ^∗T M (see [11]). Then, we have the following

Proposition 3.1. Letξ∈Γ(π^∗T M)andX ∈Γ(TT M˚ ). IfH ∼=π^∗T M then, using π_∗

( δ δxⁱ

)

= ∂

∂xⁱ, π_∗ (

F ∂

∂yⁱ )

= 0, (3.3)

one get locallyξ=ξ^{i ∂}_∂xi with ξⁱ= (X^h)ⁱ and ifV ∼=π^∗T M and, using θ

( δ δxⁱ

)

= 0, θ (

F ∂

∂yⁱ )

= ∂

∂xⁱ, (3.4)

one get locallyξ=ξ^{i ∂}_∂xi with ξⁱ= (X^v)ⁱ.

Definition 3.1. Let (M, F) be C^∞ Finslerian manifold and ˚T M its slit tangent bundle. Let u be a C^∞ function on ˚T M. The h-gradient of u is a section of the pulled-back tangent bundleπ^∗T M given by

O^hu=g^ij δu δxⁱ

∂

∂x^j. (3.5)

Thev-gradient ofuis a section of the pulled-back tangent bundleπ^∗T M given by O^vu=g^ijF ∂u

∂yⁱ

∂

∂x^j. (3.6)

3.2 Divergence of a section of the bundle π

T M

Definition 3.2. For a C^∞ section ξ ∈Γ(π^∗T M), one defines the horizontal diver- gence by

div^hξ=traceg(η7−→ ∇η^hξ) (3.7)

and the vertical divergence by

div^vξ=traceg(η7−→ ∇η^vξ) (3.8)

wheregis the fundamental tensor associated withF and∇is the Chern connection.

In the basis sections{_∂x^∂i}i=1,...,n of the vector bundleπ^∗T M, one has:

div^hξ=g^ijg (

∇ ^δ

δxi

ξ, ∂

∂x^j )

and div^vξ=g^ijg (

∇F ^∂

∂yi

ξ, ∂

∂x^j )

. (3.9)

The vertical divergence ofξis aC^∞ function on ˚T M. We get div^vξ= F ∂ξⁱ

∂yⁱ . (3.10)

3.3 Laplacians of C

functions on T M ˚

Definition 3.3. Let (M, F) be a C^∞ Finslerian manifold and u∈ C^∞(˚T M). The horizontal Laplacian ∆^huofuis given by

∆^hu=div^h(O^hu) (3.11)

and the vertical Laplacian ∆^vuofuis defined by

∆^vu=div^v(O^vu).

(3.12)

Lemma 3.2. Let(M, F)be a Finslerian manifold. For every functionu∈C^∞(˚T M), we have

∆^vu=g^ij [

F ∂

∂yⁱ (

F ∂u

∂y^j )

−2g^mkAijkF ∂u

∂y^m ]

.

Then∆^v is a second-order differential operator. Furthermore ∆^v isR-linear.

Proof. The vertical Laplacian ∆^vuis locally obtained by

∆^vu=g^ijg (

∇F ^∂

∂yiO^vu, ∂

∂x^j )

. (3.13)

By the almostg-compatibility of the Chern connection (Proposition 2.2) in the vertical directions, we have

∆^vu = g^ij {

F ∂

∂yⁱ [

g (

O^vu, ∂

∂x^j )]

−g (

O^vu,∇F ^∂

∂yi

∂

∂x^j )

−2A (

θ (

F ∂

∂yⁱ )

,O^vu, ∂

∂x^j ) }

= g^ij {

F ∂

∂yⁱ [

g (

g^mkF ∂u

∂y^m

∂

∂x^k, ∂

∂x^j )]

−2A (

θ (

F ∂

∂yⁱ )

, g^mkF ∂u

∂y^m

∂

∂x^k, ∂

∂x^j ) }

= g^ij {

F ∂

∂yⁱ [

F ∂u

∂y^mδ^m_j ]

−2g^mkF ∂u

∂y^mA ( ∂

∂xⁱ, ∂

∂x^k, ∂

∂x^j )}

= g^ij∇F ^∂

∂yi∇F ^∂

∂yi

u−g^ijg^mk∇F_∂ym^∂ gij∇F ^∂

∂yk

u.

(3.14)

One can show that for anyf, g∈C^∞(˚T M) and for everyc∈R

∆^v(f+g) = ∆^vf + ∆^vg and ∆^v(cf) =c∆^vf.

That is, ∆^v isR-linear.

Remark 3.4. As explain in Remark 2.2, the operator ∆^v, defined above, acts on C^∞(˚T M) or onC^∞(SM). For homogeneousC^∞ functions of degreer6= 0 on ˚T M, we use the non-homogeneous coordinates{_∂y^∂i} to define ∆^v. We get

O^vu=g^ij ∂u

∂yⁱ

∂

∂x^j and div^vξ= ∂ξⁱ

∂yⁱ ∀u∈C^∞(˚T M), ∀ξ∈Γ(π^∗T M).

Then, in the following we will use the homogeneous coordinates {F_∂y^∂_i} for objects onSM only.

We have the following.

Corollary 3.3.Let(M, F)be a Finslerian manifold. For every functionu∈C^∞(˚T M), we have

∆^vu=g^ij ∂²u

∂yⁱ∂y^j −g^ijg^mk∂gij

∂y^m

∂u

∂y^k. (3.15)

Then∆^v is a second-order differential operator andR-linear.

Proof. The proof is similar to the proof of the Lemma 3.2.

4 Main results

4.1 Vertical divergence lemma

Let (M, F) be an n-dimensional C^∞ Finslerian manifold. Let ξ := ξ^{i ∂}_∂xi be an arbitraryC^∞ local section of the pulled-back tangent bundle π^∗T M. ξ is a tensor field of rank (1,0) on the manifold ˚T M. In order to find the divergence formula for a section ofπ^∗T M, we need a volume form on ˚T M. We consider the 2n-form

η_F := (−1)^n(n−1)2

n! ∧ⁿdω, (4.1)

with ω := _∂y^∂F_idxⁱ the Hilbert form. The η_F is a volume form on ˚T M, called the Dazord volume form of (M, F) [5].

Remark 4.1. On the sphere bundleSM, we have the (2n−1)-form, defined by η^o_F :=(−1)^n(n−1)2

(n−1)! ω∧(dω)ⁿ⁻¹, (4.2)

known also as the Dazord volume form of (M, F).

We recall the following:

Theorem 4.1.[8] LetM be an oriented compact manifold with a fixed volume element dv. For every vector fieldX onM, we have

∫

M

divXdv= 0.

(4.3)

Remark 4.2. (1) The above formula is valid for a non-compact manifold M as long asX has a compact support [8].

(2) Every sectionξ of the vector bundle π^∗T M, with identificationπ^∗T M ∼=V :=

ker(π_∗), can be considered as a vector field on ˚T M. We have the following vertical divergence lemma.

Lemma 4.2. Let (M, F) be an oriented Finslerian manifold, T M˚ the slit tangent bundle ofM andπ^∗T M the pulled-back tangent bundle by the projectionπ: ˚T M −→

M given by π(x, y) = x. Then, for every section ξ of π^∗T M which has a compact support inT M˚ , with identificationπ^∗T M ∼=V :=ker(π_∗), we have

∫

T M˚

(div^vξ)ηF = 0.

(4.4)

Proof. The proof is obtained by using the Theorem 4.1 and the Remark 4.2.

4.2 Hopf-type theorems for Finslerian manifolds

We have the following results.

Theorem 4.3. Let (M, F)be an oriented Finslerian manifold and T M˚ its slit tan- gent bundle. Then every globally defined C^∞ function u on T M˚ , whose v-gradient has compact support with ∆^vu ≥ 0 everywhere or ∆^vu ≤ 0 everywhere, must be independant of directional arguments. In particular, the slit tangent bundle of a Fins- lerian manifold has no vertically harmonic function except for functions independant of directional arguments.

Proof. Foru, f∈C^∞(˚T M) andξ∈Γ(π^∗T M) one easily show that div^v(f ξ) = f div^vξ+g(O^vf, ξ)

(4.5)

wheregis the fundamental tensor. By the Remark 3.4, it follows that div^v(fO^vu) = f∆^vu+g(O^vu,O^vf)

(4.6)

Applying the Lemma 4.2 in (4.5), we get

∫

T M˚

(f∆^vu)ηF =−

∫

T M˚

g(O^vu,O^vf)ηF. (4.7)

In particular

∫

T M˚

(u∆^vu)ηF =−

∫

T M˚

|O^vu|²gηF

(4.8)

where |O^vu|g denote the norm of O^vu induced by the fundamental tensor g of F, given by

|O^vu|²g=g^ij(∇ ^∂

∂yi

u)(∇ ^∂

∂yj

u) =g^iju;iu;j. (4.9)

By the Lemma 4.2 again we obtain, from the expression (3.12), that

∫

T M˚

(∆^vu)ηF = 0.

(4.10)

Since ∆^v is R-linear, by replacing uby−uif necessary, we may assume without loss of generality that ∆^vu ≥ 0 everywhere. Since u ∈ C²(˚T M) and ∆^vu ≥ 0 by

assumption, we conclude that ∆^vu= 0 everywhere on ˚T M, that is uis v-harmonic.

It follows that

∫

˚T M

(u∆^vu)ηF = 0.

(4.11)

The expression (4.8) becomes

−

∫

T M˚

g^ij(∇ ^∂

∂yi

u)(∇ ^∂

∂yj

u)η_F = 0.

(4.12)

Hence u;i = _∂y^∂ui vanishes identically on ˚T M and u must be independant of the

directiony.

Theorem 4.4. Let(M, F)be a closed Finslerian manifold andSMthe sphere bundle ofM. Then every globally defined function u∈C^∞(SM), with ∆^vu≥0 everywhere or∆^vu≤0 everywhere, must be independant of direction arguments. In particular, the sphere bundle of a Finslerian manifold has no v-harmonic function except for functions depending on the base point only.

Proof. A straightforward computation of the vertical divergence of the section ξ of π^∗T M overSM shows thatdiv^vξ=F^∂ξ_∂yⁱi and the proof is obtained by the Theorem

4.1 and the Lemma 4.2.

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Authors’ addresses:

Gilbert Nibaruta

Section des Math´ematiques, Ecole Normale Sup´erieure, Avenue Mwezi Gisabo, PO Box 6983 Bujumbura-Burundi.

E-mail: [email protected] Aboubacar Nibirantiza

D´epartement des Math´ematiques, Institut de P´edagogie Appliqu´ee, Universit´e du Burundi, PO Box 2523 Bujumbura-Burundi.

E-mail: [email protected] Menedore Karimumuryango

Institut des Statistique Appliqu´ees, Universit´e du Burundi, PO Box 5158 Bujumbura-Burundi.

E-mail: [email protected] Domitien Ndayirukiye

Section des Math´ematiques, Ecole Normale Sup´erieure, Avenue Mwezi Gisabo, PO Box 6983 Bujumbura-Burundi.

E-mail: [email protected]