PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 89(103) (2011), 57–68 DOI:

THE SCALAR CURVATURE OF THE TANGENT BUNDLE OF A FINSLER MANIFOLD

Aurel Bejancu and Hani Reda Farran

Communicated by Darko Milinković

Abstract. Let 𝔽^{m} = (M, F) be a Finsler manifold and G be the Sasaki– Finsler metric on the slit tangent bundle T M^{0} = T M ∖ {0} of M. We express the scalar curvature ρ̃ of the Riemannian manifold (T M^{0}, G) in terms of some geometrical objects of the Finsler manifold 𝔽^{m}. Then, we find necessary and sufficient conditions for ρ̃ to be a positively homogenenous function of degree zero with respect to the fiber coordinates of T M^{0}. Finally, we obtain characterizations of Landsberg manifolds, Berwald manifolds and Riemannian manifolds whose ρ̃ satisfies the above condition.

Introduction

The geometry of the tangent bundle T M of a Riemannian manifold (M, g) goes back to Sasaki [10], who constructed on T M a Riemannian metric G which in our days is called the Sasaki metric. Then, several papers on the interrelations between the geometries of (M, g) and (T M, G) have been published (see Gudmundsson and Kappos [6] for results and references). The extension of the study from Riemannian manifolds to Finsler manifolds is not an easy task. This is because a Finsler manifold 𝔽^{m} = (M, F) does not admit a canonical linear connection on M, that plays the role of the Levi–Civita connection on a Riemannian manifold. Recently, the first author (cf. [3]) has initiated a study of the interrelations between the geometries of both the tangent bundle and indicatrix bundle of a Finsler manifold on one side, and the geometry of the manifold itself, on the other side. The main tool in the study was the Vrănceanu connection induced by the Levi–Civita connection on (T M^{0}, G), where G is the Sasaki–Finsler metric on T M^{0}.

We study the geometry of a Finsler manifold 𝔽^{m} = (M, F) under the assumption that the scalar curvature ρ̃ of (T M^{0}, G) is a positively homogeneous function of degree zero with respect to the fiber coordinates (y^{i}) of T M^{0}. In the first part we present some geometric objects from the geometries of 𝔽^{m} and (T M^{0}, G) and following [3] we give some structure equations which relate the curvature tensor fields of the Levi–Civita connection and the Vrănceanu connection on (T M^{0}, G). In the second part we express ρ̃ in terms of some geometric objects of the Finsler manifold 𝔽^{m} (cf. Theorem 2.1) and obtain necessary and sufficient conditions for ρ̃ to be positively homogeneous of degree zero with respect to (y^{i}) (cf. Theorem 2.2). In particular, we prove that such an 𝔽^{m} is locally Minkowskian, provided M is a compact connected boundaryless manifold (cf. Corollary 2.1). Finally, we show that if 𝔽^{m} is a Berwald manifold (cf. Corollary 2.4) or a Riemannian manifold (cf. Corollary 2.5) and ρ̃ satisfies the above condition, then 𝔽^{m} must be locally Minkowskian or locally Euclidean, respectively. In case of a Riemannian manifold, our result improves a well known result of Musso–Tricerri [9].

2010 Mathematics Subject Classification: Primary 53C60, 53C15.

Key words and phrases: Berwald manifold, Finsler manifold, Landsberg manifold, Riemannian manifold, scalar curvature, tangent bundle.

1. Preliminaries

Let 𝔽^{m} = (M, F) be an m-dimensional Finsler manifold, where F is the fundamental function of 𝔽^{m} that is supposed to be of class C^{∞} on the slit tangent bundle T M^{0} = T M ∖ {0}. Denote by (x^{i}, y^{i}), i ∈ {1, …, m}, the local coordinates on T M, where (x^{i}) are the local coordinates of a point x ∈ M and (y^{i}) are the coordinates of a vector y ∈ T_{x}M. Then, F is positively homogeneous of degree 1 with respect to (y^{i}) and the functions

g_{ij} = \frac{1}{2} \frac{∂^{2}F^{2}}{∂y^{i}∂y^{j}},

define a symmetric Finsler tensor field of type (0, 2) on T M^{0}. We suppose that the m × m matrix [g_{ij}] is positive definite and denote its inverse by [g^{ij}].

Next, we consider the vertical bundle V T M^{0} over T M^{0}, which is the kernel of the differential of the projection map Π : T M^{0} → M. Denote by Γ(V T M^{0}) the F(T M^{0})-module of sections of V T M^{0}, where F(T M^{0}) is the algebra of smooth functions on T M^{0}. The same notation will be used for any other vector bundle. Locally, Γ(V T M^{0}) is spanned by the natural vector fields {∂/∂y^{1}, …, ∂/∂y^{m}}. Then, we define the vector fields

\frac{δ}{δx^{i}} = \frac{∂}{∂x^{i}} − G_{i}^{j} \frac{∂}{∂y^{j}} , i ∈ \{1, …, m\},

where we put

G_{i}^{j} = \frac{∂G^{j}}{∂y^{i}} and G^{j} = \frac{1}{4} g^{jk} \{\frac{∂^{2}F^{2}}{∂y^{k}∂x^{i}} y^{i} − \frac{∂F^{2}}{∂x^{k}} \} .

Thus, we obtain the horizontal bundle HT M^{0} over T M^{0}, which is locally spanned by {δ/δx^{1}, …, δ/δx^{m}}. Moreover, we have the decomposition

T T M^{0} = HT M^{0} ⊕ V T M^{0},

which enables us to define the Sasaki–Finsler metric G on T M^{0} as follows (cf. Bejancu–Farran [4, p. 35])

(1.1)  G (\frac{δ}{δx^{j}} , \frac{δ}{δx^{i}} ) = G (\frac{∂}{∂y^{j}} , \frac{∂}{∂y^{i}} ) = g_{ij}(x, y), G (\frac{δ}{δx^{j}} , \frac{∂}{∂y^{i}} ) = 0.

THE SCALAR CURVATURE OF THE TANGENT BUNDLE OF A FINSLER MANIFOLD 59

Now, we define some geometric objects of Finsler type on T M^{0}. First, we express the Lie brackets of the above vector fields as follows:

[\frac{δ}{δx^{i}} , \frac{δ}{δx^{j}} ] = R^{k}_{ij} \frac{∂}{∂y^{k}} , [\frac{δ}{δx^{i}} , \frac{∂}{∂y^{j}} ] = G_{i}^{k}_{j} \frac{∂}{∂y^{k}} ,

where we put

R^{k}_{ij} = \frac{δG_{i}^{k}}{δx^{j}} − \frac{δG_{j}^{k}}{δx^{i}} , G_{i}^{k}_{j} = \frac{∂G_{j}^{k}}{∂y^{i}} .

If R^{k}_{ij} = 0 for all i, j, k ∈ {1, …, m}, we say that 𝔽^{m} is a flat Finsler manifold. This name is justified by the fact that in this case the flag curvature of 𝔽^{m} vanishes identically on T M^{0}. Also, the functions

F_{i}^{k}_{j} = \frac{1}{2} g^{kh} \{\frac{δg_{hi}}{δx^{j}} + \frac{δg_{hj}}{δx^{i}} − \frac{δg_{ij}}{δx^{h}} \} ,

represent the local coefficients of Chern–Rund connection. Then, we define a Finsler tensor field of type (1, 2) whose local components are given by B_{i}^{k}_{j} = F_{i}^{k}_{j} − G_{i}^{k}_{j}. Finally, the Cartan tensor field is given by its local components

C_{i}^{k}_{j} = \frac{1}{2} g^{kh} \frac{∂g_{ij}}{∂y^{h}} ·

Next, we denote by h and v the projection morphisms of T T M^{0} on HT M^{0} and V T M^{0}, respectively. Then, by using the above Finsler tensor fields R^{k}_{ij}, C_{i}^{k}_{j} and B_{i}^{k}_{j} we define the following adapted tensor fields:

(1.2) R : Γ(HT M^{0}) × Γ(HT M^{0}) → Γ(V T M^{0}), R(hX, hY) = R^{k}_{ij}Y^{i}X^{j} \frac{∂}{∂y^{k}}

(1.3) C : Γ(HT M^{0}) × Γ(HT M^{0}) → Γ(V T M^{0}), C(hX, hY) = C_{i}^{k}_{j}Y^{i}X^{j} \frac{∂}{∂y^{k}} ,

(1.4) B : Γ(V T M^{0}) × Γ(V T M^{0}) → Γ(HT M^{0}), B(vU, vW) = B_{i}^{k}_{j}W^{i}U^{j} \frac{δ}{δx^{k}} ,

where we set

hX = X^{j} \frac{δ}{δx^{j}} , hY = Y^{i} \frac{δ}{δx^{i}} , vU = U^{j} \frac{∂}{∂y^{j}} , vW = W^{i} \frac{∂}{∂y^{i}} ·

For each of the above tensor fields R, C and B we define a twin (denoted by the same symbol) as follows:

(1.5)  \begin{matrix}R : Γ(HT M^{0}) × Γ(V T M^{0}) → Γ(HT M^{0}), \\ g(R(hX, vY), hZ) = G(R(hX, hZ), vY), \end{matrix}

(1.6)  \begin{matrix}C : Γ(HT M^{0}) × Γ(V T M^{0}) → Γ(HT M^{0}), \\ G(C(hX, vY), hZ) = G(C(hX, hZ), vY), \end{matrix}

(1.7)  \begin{matrix}B : Γ(HT M^{0}) × Γ(V T M^{0}) → Γ(V T M^{0}), \\ G(B(hX, vY), vZ) = G(B(vY, vZ), hX). \end{matrix}

Locally, we have the following formulas:

(1.8)  \begin{matrix}(a) R (\frac{δ}{δx^{j}} , \frac{δ}{δx^{i}} ) = R^{k}_{ij} \frac{∂}{∂y^{k}} , (b) C (\frac{δ}{δx^{j}} , \frac{δ}{δx^{i}} ) = C_{i}^{k}_{j} \frac{∂}{∂y^{k}} , \\ (c) B (\frac{∂}{∂y^{j}} , \frac{∂}{∂y^{i}} ) = B_{i}^{k}_{j} \frac{δ}{δx^{k}} , \end{matrix}

(1.9)  \begin{matrix}(a) R (\frac{δ}{δx^{j}} , \frac{∂}{∂y^{i}} ) = \overline{R}^{k}_{ij} \frac{δ}{δx^{k}} , (b) C (\frac{δ}{δx^{j}} , \frac{∂}{∂y^{i}} ) = \overline{C}_{i}^{k}_{j} \frac{δ}{δx^{k}} , \\ (c) B (\frac{δ}{δx^{j}} , \frac{∂}{∂y^{i}} ) = \overline{B}_{i}^{k}_{j} \frac{∂}{∂y^{k}} , \end{matrix}

(1.10)  (a) \overline{R}^{k}_{ij} = g_{ih} R^{h}_{tj} g^{tk}, (b) \overline{C}_{i}^{k}_{j} = C_{i}^{k}_{j}, (c) \overline{B}_{i}^{k}_{j} = B_{i}^{k}_{j}.

Now, let ˜∇ be the Levi–Civita connection on (T M^{0}, G) and ∇ be the Vrănceanu connection induced by ˜∇ given by (cf. Ianuş [7])

∇_{X}Y = v ˜∇_{vX}vY + h ˜∇_{hX}hY + v[hX, vY] + h[vX, hY].

It is important to note that the Vrănceanu connection is locally given by the local coefficients of the classical Finsler connections as follows:

(1.11)  \begin{matrix}∇_{\frac{δ}{δx^{j}}} \frac{δ}{δx^{i}} = F_{i}^{k}_{j} \frac{δ}{δx^{k}} , & ∇_{\frac{∂}{∂y^{j}}}  \frac{∂}{∂y^{i}} = C_{i}^{k}_{j} \frac{∂}{∂y^{k}} , \\ ∇_{\frac{∂}{∂y^{j}}}  \frac{δ}{δx^{i}} = 0, & ∇_{\frac{δ}{δx^{j}}}  \frac{∂}{∂y^{i}} = G_{i}^{k}_{j} \frac{∂}{∂y^{k}} · \end{matrix}

Moreover, the curvature tensor field R̃ of the Levi–Civita connection ˜∇ is completely determined by the curvature tensor field R of the Vrănceanu connection on (T M^{0}, G) and the adapted tensor fields R, C and B (cf. Bejancu [3]). We recall here only the following relations:

(1.12)

R̃(hX, hY, hZ) = R(hX, hY, hZ) + B(hZ, R(hX, hY))

+ C(hZ, R(hX, hY)) + \frac{1}{2} R(hZ, R(hX, hY))

− A_{(hX,hY)}\{(∇_{hX}C)(hY, hZ) + \frac{1}{2}(∇_{hX}R)(hY, hZ)

+ B(hX, C(hY, hZ)) + \frac{1}{2} B(hX, R(hY, hZ))

+ C(hX, C(hY, hZ)) + \frac{1}{2} C(hX, R(hY, hZ))

+ \frac{1}{2} R(hX, C(hY, hZ)) + \frac{1}{4} R(hX, R(hY, hZ))\},

(1.13) \begin{matrix}R̃(hX, vY, vZ) = R(hX, vY, vZ) − (∇_{hX}B) (vY, vZ) \\ − (∇_{vY} B) (hX, vZ) − (∇_{vY} C) (hX, vZ) − \frac{1}{2} (∇_{vY} R) (hX, vZ) \\ + C(hX, B(vY, vZ)) + \frac{1}{2} R(hX, B(vY, vZ)) + B(vY, B(hX, vZ)) \\ − C(C(hX, vZ), vY) − \frac{1}{2} C(R(hX, vZ), vY) \\ −\frac{1}{2} R(C(hX, vZ), vY) − \frac{1}{4} R(R(hX, vZ), vY) \\ − B(C(hX, vZ), vY) − \frac{1}{2} B(R(hX, vZ), vY), \end{matrix}

(1.14) \begin{matrix}R̃(vX, vY, vZ) = R(vX, vY, vZ) − A_{(vX,vY)}\{(∇_{vX}B)(vY, vZ) \\ + C(B(vY, vZ), vX) + \frac{1}{2} R(B(vY, vZ), vX) + B(B(vY, vZ), vX)\}, \end{matrix}

where A_{(hX,hY)} means that in the expression that follows this symbol we interchange hX and hY, and then subtract, as in the following formula

A_{(hX,hY)}{f(hX, hY)} = f(hX, hY) − f(hY, hX).

In a similar way, we use the symbol A_{(vX,vY)}. Finally, we present some local components of the curvature tensor field of the Vrănceanu connection on (T M^{0}, G):

(1.15)  \begin{matrix}(a) R(\frac{δ}{δx^{k}} , \frac{δ}{δx^{j}} )\frac{δ}{δx^{i}} = K_{i}^{h}_{jk} \frac{δ}{δx^{h}} , \\ (b) R(\frac{∂}{∂y^{k}} , \frac{∂}{∂y^{j}} )\frac{∂}{∂y^{i}} = S_{i}^{h}_{jk} \frac{∂}{∂y^{h}} \end{matrix}

where we set

(1.16)  \begin{aligned}(a) K_{i}^{h}_{jk} & = \frac{δF_{i}^{h}_{j}}{δx^{k}} − \frac{δF_{i}^{h}_{k}}{δx^{j}} + F_{i}^{t}_{j} F_{t}^{h}_{k} − F_{i}^{t}_{k} F_{t}^{h}_{j}, \\ (b) S_{i}^{h}_{jk} & = \frac{∂C_{i}^{h}_{j}}{∂y^{k}} − \frac{∂C_{i}^{h}_{k}}{∂y^{j}} + C_{i}^{t}_{j} C_{t}^{h}_{k} − C_{i}^{t}_{k} C_{t}^{h}_{j}. \end{aligned}

We note that (1.16a) and (1.16b) give the local components of the hh-curvature and vv-curvature tensor fields of the Chern–Rund connection and Cartan connection, respectively.

2. Scalar curvature of (T M^{0}, G)

Let 𝔽^{m} = (M, F) be a Finsler manifold and (T M^{0}, G) be its slit tangent bundle endowed with the Sasaki–Finsler metric G given by (1.1). Consider the local orthonormal fields of frames {H_{a}} and {V_{a}}, such that H_{a} ∈ Γ(HT M^{0}) and V_{a} ∈ Γ(V T M^{0}) for any a ∈ {1, …, m}. Next, we set

(2.1)  H_{a} = H_{a}^{i} \frac{δ}{δx^{i}} and V_{a} = V_{a}^{i} \frac{∂}{∂y^{i}} ·

Then, by using (1.1), we deduce that the inverse matrix of [g_{ij}] has the entries given by

(2.2)  g^{ij} = ∑_{a=1}^{m} H_{a}^{i} H_{a}^{j} = ∑_{a=1}^{m} V_{a}^{i} V_{a}^{j}, i, j ∈ {1, …, m}.

Now we denote by ρ̃ the scalar curvature of the Riemannian manifold (T M^{0}, G). As {H_{a}, V_{a}}, a ∈ {1, …, m}, is a local orthonormal frame field on T M^{0} with respect to G, we have

(2.3)  ρ̃ = α + 2β + γ,

where we put

(a) α = ∑_{a,b=1}^{m} {G(R̃(H_{a}, H_{b})H_{b}, H_{a})},

(2.4)  (b) β = ∑_{a,b=1}^{m} {G(R̃(H_{a}, V_{b})V_{b}, H_{a})},

(c) γ = ∑_{a,b=1}^{m} {G(R̃(V_{a}, V_{b})V_{b}, V_{a})}.

In what follows we will express the above three functions α, β, γ in terms of the local components of some important Finsler tensor fields.

First, by using (1.5) and (1.6) and taking into account that R and C are skewsymmetric and symmetric adapted tensor fields respectively, we obtain

(a) G(C(H_{a}, C(H_{b}, H_{b})), H_{a}) = G(C(H_{a}, H_{a}), C(H_{b}, H_{b})),
(b) G(C(H_{b}, C(H_{a}, H_{b})), H_{a}) = ∥C(H_{a}, H_{b})∥^{2},
(c) G(C(H_{b}, R(H_{a}, H_{b})), H_{a}) = −G(R(H_{b}, C(H_{a}, H_{b})), H_{a})

= G(C(H_{a}, H_{b}), R(H_{a}, H_{b})), (2.5)

(d) G(R(H_{a}, C(H_{b}, H_{b})), H_{a}) = G(R(H_{a}, H_{a}), C(H_{b}, H_{b})) = 0,
(e) G(R(H_{b}, R(H_{a}, H_{b})), H_{a}) = −∥R(H_{a}, H_{b})∥^{2},
(f)  ∑_{a,b=1}^{m} {G(C(H_{a}, H_{b}), R(H_{a}, H_{b}))} = 0,

where the norm ∥ · ∥ is taken with respect to G. Then, by direct calculations using (2.4a), (1.12) and (2.5), we deduce that

α = ∑_{a,b=1}^{m} {G(hR̃(H_{a}, H_{b})H_{b}, H_{a})}

(2.6)  = ∑_{a,b=1}^{m} \{G(R(H_{a}, H_{b})H_{b}, H_{a}) − \frac{3}{4}∥R(H_{a}, H_{b})∥^{2} + ∥C(H_{a}, H_{b})∥^{2}
− G(C(H_{a}, H_{a}), C(H_{b}, H_{b}))}.

THE SCALAR CURVATURE OF THE TANGENT BUNDLE OF A FINSLER MANIFOLD 63

Next, by using (1.5), (1.6) and (1.7), we obtain

(a) G(B(V_{b}, B(H_{a}, V_{b})), H_{a}) = ∥B(H_{a}, V_{b})∥^{2},
(b) G(C(C(H_{a}, V_{b}), V_{b}), H_{a}) = ∥C(H_{a}, V_{b})∥^{2},
(2.7)  (c) G(C(R(H_{a}, V_{b}), V_{b}), H_{a}) + G(R(C(H_{a}, V_{b}), V_{b}), H_{a}) = 0,
(d) G(R(R(H_{a}, V_{b}), V_{b}), H_{a}) = −∥R(H_{a}, V_{b})∥^{2}.

Then, taking into account (2.4b), (1.13) and (2.7), we infer that

(2.8) \begin{aligned}β & = ∑_{a,b=1}^{m} \{∥B(H_{a}, V_{b})∥^{2} − ∥C(H_{a}, V_{b})∥^{2} + \frac{1}{4} ∥R(H_{a}, V_{b})∥^{2} \\ & − G((∇_{H_{a}}B)(V_{b}, V_{b}) + (∇_{V_{b}}C)(H_{a}, V_{b}) + \frac{1}{2}(∇_{V_{b}}R)(H_{a}, V_{b}), H_{a})\}. \end{aligned}

Now, as a consequence of (1.7), we obtain

(2.9)  \begin{matrix}(a) & G(B(B(V_{b}, V_{b}), V_{a}), V_{a}) = G(B(V_{a}, V_{a}), B(V_{b}, V_{b})) \\ (b) & G(B(B(V_{a}, V_{b}), V_{b}), V_{a}) = ∥B(V_{a}, V_{b})∥^{2}. \end{matrix}

Then, by using (2.4c), (1.14) and (2.9), we deduce that

(2.10) γ = ∑_{a,b=1}^{m} {G(R(V_{a}, V_{b})V_{b}, V_{a}) + ∥B(V_{a}, V_{b})∥^{2} − G(B(V_{a}, V_{a}), B(V_{b}, V_{b}))}.

Also, by using (2.1), (2.2), (1.8), (1.9) and (1.10), we obtain

(2.11) \begin{aligned}(a) ∑_{a,b=1}^{m} ∥R(H_{a}, V_{b})∥^{2} & = ∑_{a,b=1}^{m} ∥R(H_{a}, H_{b})∥^{2} = g^{ik} g^{jh} g_{st} R^{s}_{ij} R^{t}_{kh}, \\ (b) ∑_{a,b=1}^{m} ∥C(H_{a}, V_{b})∥^{2} & = ∑_{a,b=1}^{m} ∥C(H_{a}, H_{b})∥^{2} = g^{ik} g^{jh} g_{st} C_{i}^{s}_{j} C_{k}^{t}_{h}, \\ (c) ∑_{a,b=1}^{m} ∥B(H_{a}, V_{b})∥^{2} & = ∑_{a,b=1}^{m} ∥B(V_{a}, V_{b})∥^{2} = g^{ik} g^{jh} g_{st} B_{i}^{s}_{j} B_{k}^{t}_{h}. \end{aligned}

Finally, by using (2.3), (2.6), (2.8), (2.10) and (2.11), we deduce that

(2.12) \begin{matrix}ρ̃ = ∑_{a,b=1}^{m}\{ & G(R(H_{a}, H_{b})H_{b}, H_{a}) + G(R(V_{a}, V_{b})V_{b}, V_{a}) \\ − \frac{1}{4}∥R(H_{a}, H_{b})∥^{2} − ∥C(H_{a}, H_{b})∥^{2} + 3∥B(V_{a}, V_{b})∥^{2} \\ − G(C(H_{a}, H_{a}), C(H_{b}, H_{b})) − G(B(V_{a}, V_{a}), B(V_{b}, V_{b})) \\ −2G((∇_{H_{a}} & B)(V_{b}, V_{b}) + (∇_{V_{b}}C)(H_{a}, V_{b}) + \frac{1}{2} (∇_{V_{b}}R)(H_{a}, V_{b}), H_{a})\}. \end{matrix}

Next, we want to express the scalar curvature of (T M^{0}, G) in terms of some geometric objects of Finsler type of 𝔽^{m}. First, by using (2.1), (2.2), (1.15), (1.8b) and (1.8c), we deduce that

(2.13)  \begin{matrix}(a) & ∑_{a,b=1}^{m} \{G(R(H_{a}, H_{b})H_{b}, H_{a})\} = g^{ik} g^{jh} K_{ijkh}, \\ (b) & ∑_{a,b=1}^{m} \{G(R(V_{a}, V_{b})V_{b}, V_{a})\} = g^{ik} g^{jh} S_{ijkh}, \\ (c) & ∑_{a,b=1}^{m} \{G(C(H_{a}, H_{a}), C(H_{b}, H_{b}))\} = g^{ik} g^{jh} g_{st} C_{i}^{s}_{k} C_{j}^{t}_{h}, \\ (d) & ∑_{a,b=1}^{m} \{G(B(V_{a}, V_{a}), B(V_{b}, V_{b}))\} = g^{ik} g^{jh} g_{st} B_{i}^{s}_{k} B_{j}^{t}_{h}. \end{matrix}

Then, by using (2.1), (2.2) and (1.11), we obtain

(2.14)  \begin{matrix}(a) & ∑_{a,b=1}^{m} \{G((∇_{H_{a}}B)(V_{b}, V_{b}), H_{a})\} = g^{jh} B_{j}^{i}_{h|i}, \\ (b) & ∑_{a,b=1}^{m} \{G((∇_{V_{b}}C)(H_{a}, V_{b}), H_{a})\} = g^{jh} C_{h}^{i}_{i∥j}, \\ (c) & ∑_{a,b=1}^{m} \{G((∇_{V_{b}}R)(H_{a}, V_{b}), H_{a})\} = g^{jh} \overline{R}^{i}_{hi∥j}, \end{matrix}

where the covariant derivatives on the right side are defined by the Vrănceanu connection as follows

(2.15)  \begin{matrix}(a) & B_{j}^{i}_{h|i} = \frac{δB_{j}^{i}_{h}}{δx^{i}} + B_{j}^{k}_{h} F_{k}^{i}_{i} − B_{k}^{i}_{h} G_{j}^{k}_{i} − B_{j}^{i}_{k} G_{h}^{k}_{i}, \\ (b) & C_{h}^{i}_{i∥j} = \frac{∂C_{h}^{i}_{i}}{∂y^{j}} − C_{k}^{i}_{i} C_{h}^{k}_{j}, \\ (c) & \overline{R}_{h}^{i}_{i∥j} = \frac{∂\overline{R}^{i}_{hi}}{∂y^{j}} − \overline{R}^{i}_{ki} C_{h}^{k}_{j}. \end{matrix}

Thus, by using (2.11), (2.13) and (2.14) into (2.12), we deduce that the scalar curvature of (T M^{0}, G) is given by

(2.16)  \begin{matrix}ρ̃ = g^{ik} g^{jh}\{ & K_{ijkh} + S_{ijkh} − \frac{1}{4} g_{st} R^{s}_{ij} R^{t}_{kh} − g_{st} C_{i}^{s}_{j} C_{k}^{t}_{h} \\ + 3g_{st} B_{i}^{s}_{j} B_{k}^{t}_{h} − g_{st} C_{i}^{s}_{k} C_{j}^{t}_{h} − g_{st} B_{i}^{s}_{k} B_{j}^{t}_{h}\} \\ − 2g^{jh}\{ & B_{j}^{i}_{h|i} + C_{h}^{i}_{i∥j} + \frac{1}{2} \overline{R}^{i}_{hi∥j}\}. \end{matrix}

An interesting formula for S_{ijkh} was given by Matsumoto [8, p. 114]:

(2.17)  S_{ijkh} = g_{st}{C_{i}^{s}_{h} C_{j}^{t}_{k} − C_{i}^{s}_{k} C_{j}^{t}_{h}}.

Then, by direct calculations, using (2.17) and (2.15b), we obtain

(2.18) g^{ik} g^{jh}\{S_{ijkh} − g_{st} C_{i}^{s}_{j} C_{k}^{t}_{h} − g_{st} C_{i}^{s}_{k} C_{j}^{t}_{h}\} − 2g^{jh} C_{h}^{i}_{i∥j} = −2g^{jh} \frac{∂C_{h}}{∂y^{j}} ,

where we put

(2.19)  C_{h} = C_{h}^{i}_{i} = g^{ki} C_{hki}.

Taking into account of (2.18) into (2.16), we can state the following.

Theorem 2.1. Let 𝔽^{m} = (M, F) be a Finsler manifold. Then, the scalar curvature ρ̃ of the Riemannian manifold (T M^{0}, G) is given by

(2.20) \begin{matrix}ρ̃ = g^{ik} g^{jh}\{ & K_{ijkh} + 3g_{st} B_{i}^{s}_{j} B_{k}^{t}_{h} − g_{st} B_{i}^{s}_{k} B_{j}^{t}_{h} − \frac{1}{4} g_{st}R^{s}_{ij} R^{t}_{kh}\} \\ − 2g^{jh}\{ & B_{j}^{i}_{h|i} + \frac{∂C_{h}}{∂y^{j}} + \frac{1}{2} \overline{R}^{i}_{hi∥j}\}. \end{matrix}

Next, following Matsumoto [8, p. 176], we call

(2.21)  C^{i} = g^{ih} C_{h},

the torsion vector field of 𝔽^{m}. Then, we can prove the following.

Theorem 2.2. Let 𝔽^{m} = (M, F) be a Finsler manifold. Then, the scalar curvature of (T M^{0}, G) is a positively homogeneous function of degree zero with respect to (y^{i}) if and only if the following conditions are satisfied:

(i) 𝔽^{m} is a flat Finsler manifold.

(ii) The torsion vector field of 𝔽^{m} satisfies

(2.22)  \frac{∂C^{i}}{∂y^{i}} + 2g_{jk} C^{j} C^{k} = 0.

Proof. First, we express (2.20) as follows

(2.23)  ρ̃ = A + B + C,

where

(2.24)  \begin{aligned}(a) & A = g^{ik} g^{jh}\{K_{ijkh} + 3g_{st} B_{i}^{s}_{j} B_{k}^{t}_{h} − g_{st} B_{i}^{s}_{k} B_{j}^{t}_{h}\} \\ & −2g^{jh}\{B_{j}^{i}_{h|i} + \frac{1}{2} \overline{R}^{i}_{hi∥j}\}, \\ (b) & B = −\frac{1}{4} g^{ik} g^{jh} g_{st} R^{s}_{ij} R^{t}_{kh}, \\ (c) & C = −2g^{jh} \frac{∂C_{h}}{∂y^{j}} · \end{aligned}

By the homogeneity properties of the functions on the right side of (2.24), we conclude that A, B and C are positively homogeneous functions of degrees 0, 1 and −2, respectively. Then, from (2.23) and (2.24), we deduce that ρ̃ is positively homogeneous of degree 0 if and only if we have

(2.25)  (a) g^{ik} g^{jh} g_{st} R^{s}_{ij} R^{t}_{kh} = 0,  (b) g^{jh} \frac{∂C_{h}}{∂y^{j}} = 0.

Clearly, (2.25a) holds if and only if R^{k}_{ij} = 0, for each i, j, k ∈ {1, …, m}, that is, 𝔽^{m} is a flat Finsler manifold. On the other hand, by using (2.21) and (2.19), we deduce that

g^{jh} \frac{∂C_{h}}{∂y^{j}} = g^{jh} \{\frac{∂g_{hk}}{∂y^{j}} C^{k} + g_{hk} \frac{∂C^{k}}{∂y^{j}} \}

= \frac{∂C^{j}}{∂y^{j}} + 2g^{jh} C_{hkj} C^{k} = \frac{∂C^{j}}{∂y^{j}} + 2g_{kh} C^{k} C^{h}.

Thus, (2.25b) is equivalent to (2.22). This completes the proof of the theorem. □

Next, we recall the following results on the geometry of 𝔽^{m}.

Theorem 2.3 (Akbar–Zadeh [1]). Let 𝔽^{m} = (M, F) be a compact connected boundaryless flat Finsler manifold. Then, 𝔽^{m} is locally Minkowskian.

Theorem 2.4 (Deicke [5]). Let 𝔽^{m} = (M, F) be a Finsler manifold such that F is positive and C^{4}-differentiable for any nonzero (y^{i}). If the torsion vector field vanishes on M, then 𝔽^{m} must be Riemannian.

Now, by combining Theorems 2.2 and 2.3, we obtain the following:

Corollary 2.1. Let 𝔽^{m} = (M, F) be a be a compact connected boundaryless Finsler manifold. If the scalar curvature of (T M^{0}, G) is a positively homogeneous function of degree zero with respect to (y^{i}), then 𝔽^{m} is locally Minkowskian.

Also, we can prove the following.

Corollary 2.2. Let 𝔽^{m} = (M, F) be a Finsler manifold such that F is positive and C^{4}-differentiable for any nonzero (y^{i}). Suppose the torsion vector field of 𝔽^{m} satisfies

(2.26)  trace [\frac{∂C^{i}}{∂y^{j}} ] ⩾ 0.

Then the scalar curvature of (T M^{0}, G) is positively homogeneous of degree 0 with respect to (y^{i}) if and only if 𝔽^{m} is locally Euclidean.

Proof. If 𝔽^{m} is locally Euclidean, then both the flag curvature and the torsion vector field of 𝔽^{m} vanish on M, and by Theorem 2.2 we conclude that ρ̃ is positively homogeneous of degree 0. Conversely, suppose that ρ̃ is positively homogeneous of degree 0. Then, from (2.26) and (2.22), we deduce that the torsion vector field of 𝔽^{m} vanishes on M. Thus, by Theorem 2.4, we conclude that 𝔽^{m} must be Riemannian. Finally, from Theorem 2.2 we see that 𝔽^{m} is a flat Finsler manifold. Hence, F^{m} is locally Euclidean. □

Corollary 2.3. Let 𝔽^{m} = (M, F) be a Landsberg manifold. If the scalar curvature of (T M^{0}, G) is a positively homogeneous function of degree 0 with respect to (y^{i}), then it vanishes on T M.

Proof. Since B_{j}^{i}_{k} = 0 for all i, j, k ∈ {1, …, m}, by (2.23) and (2.24) we deduce that ρ̃ = g^{ik} g^{jh} K_{ijkh}. Also, by assertion (i) of Theorem 2.2, we conclude that 𝔽^{m} is of flag curvature λ = 0. On the other hand, the hh-curvature tensor of the Chern–Rund connection for a Landsberg manifold of constant curvature λ is given by (cf. Bao–Chern–Shen [2, p. 314])

K_{ijkh} = λ(g_{jk} g_{ih} − g_{ik} g_{jh}).

Thus, ρ̃ vanishes on T M. □

Corollary 2.4. Let 𝔽^{m} = (M, F) be a Berwald manifold. Suppose that the scalar curvature ρ̃ of (T M^{0}, G) is a positively homogeneous function of degree 0 with respect to (y^{i}). Then ρ̃ = 0, and 𝔽^{m} is locally Minkowskian.

Proof. As 𝔽^{m} is a Landsberg manifold, we apply Corollary 2.3 and obtain ρ̃ = 0. Then, by assertion (iv) of Theorem 8.5 of Bejancu–Farran [4, p. 65], we deduce that the hv-curvature tensor field F_{ijkh} of the Chern–Rund connection vanishes on M. Finally, from the proof of Corollary 2.3, we deduce that K_{ijkh} = 0. Hence, by assertion (iv) of Theorem 8.6 of Bejancu–Farran [4, p. 66], we conclude that 𝔽^{m} is locally Minkowskian. □

Corollary 2.5. Let 𝔽^{m} = (M, F) be a Riemannian manifold and T M be equipped with the Sasaki metric G. Suppose that the scalar curvature ρ̃ of (T M^{0}, G) is a positively homogeneous function of degree 0 with respect to (y^{i}). Then ρ̃ = 0, and 𝔽^{m} is locally Euclidean.

Proof. By Corollary 2.4 we have ρ̃ = 0 and K_{ijkh} = 0. As in this case K_{ijkh} are the local components of the curvature tensor field of the Levi–Civita connection on (M, g), we conclude that (M, g) is locally Euclidean. □

Finally, we note that Corollary 2.5 improves some well-known results of Musso– Tricerri [9] and Yano–Okubo [11] which state the following:

If the scalar curvature of (T M, G) is constant, then (M, g) is locally Euclidean. and

If the scalar curvature of (T M, G) vanishes, then (M, g) is locally Euclidean.

References

1. H. Akbar–Zadeh, Sur les espaces de Finsler à courbures sectionelles constantes, Bull. Acad. Roy. Bel. Cl. Sci. (5) 74 (1988), 281–322.

2. D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry, Springer-Verlag, New York, 2000.

3. A. Bejancu, Tangent bundle and indicatrix bundle of a Finsler manifold, Kodai Math. J. 31 (2008), 272–306.

4. A. Bejancu and H. R. Farran, Geometry of pseudo-Finsler Submanifolds, Kluwer, Dordrecht, 2000.

5. A. Deicke, Über die Finsler–Räume mit A_{i} = 0, Arch. Math. 4 (1953), 45–51.

6. S. Gudmundsson and E. Kappos, On the geometry of tangent bundles, Expositiones Math. 20 (2002), 1–41.

7. S. Ianuş, Some almost product structures on manifolds with linear connections, Kodai Math. Sem. Rep. 23 (1971), 305–310.

8. M. Matsumoto, Foundations of Finsler Geometry and Special Finsler Spaces, Kaiseisha Press, Saikawa, 1986.

9. E. Musso and F. Triceri, Riemannian metrics on tangent bundles, Ann. Mat. Pura Appl. 150 (1988), 1–20.

10. S. Sasaki, On differential geometry of tangent bundles of Riemannian manifolds, Tôhoku Math. J. 10 (1958), 338–354.

11. K. Yano and T. Okubo, On tangent bundles with Sasakian metrics of Finslerian and Riemannian manifolds, Ann. Mat. Pura Appl. 87 (1970), 137–162.

Department of Mathematics and Computer Science (Received 07 04 2009) Kuwait University Kuwait and Institute of Mathematics Iasi Branch of the Romanian Academy Romania

aurel.bejancu@ku.edu.kw

Department of Mathematics and Computer Science Kuwait University Kuwait

hani.farran@ku.edu.kw