Criteria for Conformal Flatness of Finsler Spaces

Fumio Ikeda

Abstract

The concept of conformal flatness of Finsler spaces is studied by several authors ([1], [2], [3], [5], [6], [7] and [8]). Especially, Kikuchi [6] obtained a very import ant theorem by using the function L^{2}C^{2} under a certain condition. Hereafter, we shall call it the Kikuchi theorem.

The purpose of the present paper is to study some necessary and sufficient conditions for Finsler spaces to be conformally flat, which are similar to the Kikuchi theorem.

Mathematics Subject Classification: 53C60 Key words: Finsler manifold, conformal flatness

1 Preliminaries

Let F = (M, L) be an n-dimensional Finsler space endowed with fundamental function L = L(x, y), where M is an n-dimensional differentiable manifold, x = (x^{i}) is a point and y = (y^{i}) is a supporting el ement of F, respectively. Then, the following notations are used

(1.1)  \begin{matrix}g_{ij} = (∂L^{2}/∂y^{i}∂y^{j})/2, C_{ijk} = (∂g_{ij}/∂y^{k})/2, \\ C_{i} = g^{jk}C_{ijk}, C^{2} = g^{ij}C_{i}C_{j}, g^{jk} = (g_{jk})^{−1}, \\ γ_{j k}^{i} = g^{ir}(∂g_{rk}/∂x^{j} + ∂g_{rj}/∂x^{k} − ∂g_{jk}/∂x^{r})/2, \\ G^{i} = γ_{j k}^{i}y^{j}y^{k}/2, G_{j}^{i} = ∂G^{i}/∂y^{j}, \\ G_{jk}^{i} = ∂G_{j}^{i}/∂y^{k}, G_{jkl}^{i} = ∂G_{jk}^{i}/∂y^{l}, \\ F_{j k}^{i} = γ_{j k}^{i} − C_{r k}^{i}G_{j}^{r} − C_{r j}^{i}G_{k}^{r} + C_{rjk}G_{t}^{r}g^{ti}, \end{matrix}

where C_{r k}^{i} = C_{rtk}g^{ti} and C_{ijk} is called the h(hv)-torsion tensor. Hereafter, we adopt the Cartan connection CΓ = (F_{j k}^{i}, G_{j}^{i}, C_{j k}^{i}) as a Finsler connection of F.

It is well-known that if all coefficients of G_{jk}^{i} of an n-dimensional Finsler space F depend on position alone, then F is called a Berwald space, and the Berwald space is characterized by C_{ijk|l} = 0, where the symbol _{|} means the h-covariant derivative with respect to the Cartan connection CΓ. Moreover, if the fundamental function L of F depends on supporting element alone under a certain coordinate system (x^{i}), then F is called a locally Minkowski space.

Balkan Journal of Geometry and Its Applications, Vol.2, No.2, 1997, pp. 63-68 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Now we consider a change of fundamental function L to another fundamental function L^{∗} on the same underlying manifold M, then we have two Finsler spaces F = (M, L) and F^{∗} = (M, L^{∗}). If there exists a scalar function σ = σ(x) of position only such that L^{∗} = e^{σ}L, then the change is called a conformal change. We introduce the following notations that will play important roles later

l^{i} = y^{i}/L, B^{ij} = L^{2}(g^{ij} − 2l^{i}l^{j})/2, B_{k}^{ij} = ∂B^{ij}/∂y^{k},
B_{km}^{ij} = ∂B_{k}^{ij}/∂y^{m}, B_{kml}^{ij} = ∂B_{km}^{ij}/∂y^{l}.

Hashiguchi [1] proved the following two propositions: Proposition A. B^{ij}, B_{k}^{ij}, B_{km}^{ij} and B_{kml}^{ij} are invariant under any conformal change. Proposition B. Under a conformal change L^{∗} = e^{σ}L, the following relations are true

(1) g_{ij}^{∗} = e^{2σ}g_{ij}, g^{∗ij} = e^{−2σ}g^{ij},
(2) G^{∗i} = G^{∗i} − B^{ir}σ_{r}, G_{j}^{∗i} = G_{j}^{i} − B_{j}^{ir}σ_{r},
(3) G_{jk}^{∗i} = G_{jk}^{i} − B_{jk}^{ir}σ_{r}, G_{jkl}^{∗i} = G_{jkl}^{i} − B_{jkl}^{ir}σ_{r},
(4) C_{j k}^{∗ i} = C_{j k}^{i}, C_{k}^{∗} = C_{k},

where σ_{r} = ∂σ/∂x^{r}.

2 The Kikuchi theorem and a similar theorem

In this section, we shall state conformally flat Finsler spaces which are introduced by the following definition:

Definition. An n-dimensional Finsler space F = (M, L) with fundamental function L is called conformally flat, if for any point p of F, there exists a local coodinate neigborhood (U, x) containing p and a function σ(x) on U, such that the Finsler space F^{∗} = (M, L^{∗}) with fundamental function L^{∗} = e^{σ}L is a locally Minkowski space.

Kikuchi payed attension to the function L^{2}C^{2} of a Finsler space F and introduced a tensor W _{j}^{i} = (∂L^{2}C^{2}/∂y^{r})B_{j}^{ri}. Moreover, under the condition that the tensor W _{j}^{i} is regular, he defined a vector B_{j}, a conformally invariant connection M_{jk}^{i} and two conformally invariant curvature tensors M_{jkl}^{i} and H_{jkl}^{i} as follows

(2.1)  B_{j} = (L^{2}C^{2})_{|k}W _{j}^{′k}, where W _{j}^{′k} = (W _{j}^{k})^{−1},

(2.2)  M_{jk}^{i} = G_{jk}^{i} − B_{jk}^{ir}B_{r},

(2.3)

\begin{aligned}M_{jkl}^{i} & = & ∂M_{jk}^{i}/∂x^{l} − (∂M_{jk}^{i}/∂y^{r})M_{lt}^{r}y^{t} − ∂M_{jl}^{i}/∂x^{k} + (∂M_{jl}^{i}/∂y^{r})M_{kt}^{r}y^{t} \\ & + & M_{jk}^{r}M_{rl}^{i} − M_{jl}^{r}M_{rk}^{i}, \end{aligned}

(2.4)  H_{jkl}^{i} = G_{jkl}^{i} − B_{jkl}^{ir}B_{r}.

The Kikuchi Theorem [6]. Let F = (M, L) be an n-dimensional Finsler space for which W _{j}^{i} is regular. The space F is conformally flat if and only if

∂B_{i}/∂y^{j} = 0, B_{i;j} − B_{j;i} = 0, H_{jkl}^{i} = 0, M_{jkl}^{i} = 0,

where the symbol ; means the h-covariant derivative with respect to the new Finsler connection (M_{jk}^{i}, M_{jk}^{i}y^{k}).

We now consider two conditions that the function L^{2}C^{2} in the Kikuchi theorem has to satisfy. The first condition is that L^{2}C^{2} is conformally invariant and the second condition is that if F^{∗} becomes a Berwald space under a conformal change L^{∗} = e^{σ}L, then L^{∗2}C^{∗2} is h-covariant constant with respect to the Cartan connection CΓ^{∗} = (F _{j k}^{∗ i}, G_{j}^{∗i}, C_{j k}^{∗ i}) of F^{∗}. So, if there exists a certain conformally invariant function A on an n-dimensional Finsler space F and satisfies the above second condition, then we get a theorem which is similar to the Kikuchi theorem exchanging the function L^{2}C^{2} for this function A.

Since the function A is conformally invariant, that is A = A^{∗}, from (1.1), it is derived that

(2.5)  A_{|∗k}^{∗} = A_{|k} + A_{k}^{r}σ_{r},

where A_{k}^{r} = (∂A/∂y^{i})B_{k}^{ir} and the symbol_{|∗} means h-covariant derivative with respect to the Cartan connection CΓ^{∗} = (F _{j k}^{∗ i}, G_{j}^{∗i}, C_{j k}^{∗ i}) of F^{∗}. Now, we assume that A_{k}^{r} is regular tensor, then transvection of (2.5) by the reciprocal tensor A_{j}^{′r} of A_{j}^{r} yields

(2.6)  σ_{j} = A_{j} − A_{j}^{∗},

where A_{j} = −A_{|k}A_{j}^{′k}, because A_{j}^{′r} and A_{j}^{r} are both conformally invariant. Put A_{jk}^{i} = G_{jk}^{i} − B_{jk}^{ir}A_{r}, then (3) of Proposition B and (2.6) show that A_{jk}^{i} becomes a conformally invariant symmetric connection. Thus, we can get a conformally invariant curvature tensor

(2.7)  \begin{aligned}A_{jkl}^{i} & = & ∂A_{jk}^{i}/∂x^{l} − (∂A_{jk}^{i}/∂y^{r})A_{lt}^{r}y^{t} − ∂A_{jl}^{i}/∂x^{k}+ \\ & + & (∂A_{jl}^{i}/∂y^{r})A_{kt}^{r}y^{t} + A_{jk}^{r}A_{rl}^{i} − A_{jl}^{r}A_{rk}^{i}. \end{aligned}

Moreover, from (3) of Proposition B and (2.6), another conformally invariant curvature tensor A_{jkl}^{′i} is definded by

(2.8)  A_{jkl}^{′i} = G_{jkl}^{i} − B_{jkl}^{ir}A_{r}.

Therefore, using a similar proof to those of Kikuchi [6], we have

Theorem 2.1. Let F = (M, L) be an n-dimensional Finsler space. If there exists a conformal invariant function A which satisfies the condition that F^{∗} = (M, L^{∗}) is a Berwald space under a conformal change L^{∗} = e^{σ}L, then A^{∗} is h-covariant constant with respect to the Cartan connection CΓ^{∗} of F^{∗} and the tensor A_{k}^{r} = (∂A/∂y^{i})B_{k}^{ir} is regular. Then F is conformally flat if and only if

∂A_{i}/∂y^{j} = 0, A_{i;j} − A_{j;i} = 0, A_{jkl}^{i} = 0, A_{jkl}^{′i} = 0,

where the symbol ; means the h-covariant derivative with respect to the new Finsler connection (A_{jk}^{i}, A_{jk}^{i}y^{k}).

3 The conditions for a Finsler space to be conformally flat

In this section, we shall find some functions satisfying two conditions in Theorem 2.1. First, we state the h(hv)-torsion tensor C_{ijk}. From Proposition B, it is easily seen that C_{ijk}^{∗} = e^{2σ}C_{ijk} and C^{∗i} = e^{−2σ}C^{i}, where C^{i} = g^{ir}C_{r}. Thus, a function D = L^{4}C_{ijk}C^{i}C^{j}C^{k} becomes conformally invariant. On the other hand, the h-covariant derivative of the function D^{∗} vanishes, if the Finsler space F^{∗} is a Berwald space. Therefore, we have Theorem 3.1. Let F = (M, L) be an n-dimensional Finsler space with detD_{j}^{i} ≠ 0. Then, a necessary and sufficient condition for F to be conformally flat is

∂D_{i}/∂y^{j} = 0, D_{i;j} − D_{j;i} = 0, D_{jkl}^{i} = 0, D_{jkl}^{′i} = 0,

where

D = L^{4}C_{ijk}C^{i}C^{j}C^{k}, D_{k}^{r} = (∂D/∂y^{i})B_{k}^{ir},
D_{j} = −D_{|k}D_{j}^{′k}, D_{j}^{′k} = (D_{j}^{k})^{−1}, D_{jk}^{i} = G_{jk}^{i} − B_{jk}^{ir}D_{r},
D_{jkl}^{i} = ∂D_{jk}^{i}/∂x^{l} − (∂D_{jk}^{i}/∂y^{r})D_{lt}^{r}y^{t} − ∂D_{jl}^{i}/∂x^{k} + (∂D_{jl}^{i}/∂y^{r})D_{kt}^{r}y^{t}+
+ D_{jk}^{r}D_{rl}^{i} − D_{jl}^{r}D_{rk}^{i},
D_{jkl}^{′i} = G_{jkl}^{i} − B_{jkl}^{ir}D_{r}.

and the symbol ; means the h-covariant derivative with respect to the new Finsler connection (D_{jk}^{i}, D_{jk}^{i}y^{k}).

Moreover, it is evident that a function E = L^{2}C_{ijk}C^{ijk} satisfies two conditions in Theorem 2.1. Therefore, we have Theorem 3.2. Let F = (M, L) be an n-dimensional Finsler space with detE_{j}^{i} ≠ 0. Then, a necessary and sufficient condition for F to be conformally flat is

∂E_{i}/∂y^{j} = 0, E_{i;j} − E_{j;i} = 0, E_{jkl}^{i} = 0, E_{jkl}^{′i} = 0,

where

E = L^{4}C_{ijk}C^{ijk}, E_{k}^{r} = (∂E/∂y^{i})B_{k}^{ir},
E_{j} = −E_{|k}D_{j}^{′k}, E_{j}^{′k} = (E_{j}^{k})^{−1}, E_{jk}^{i} = G_{jk}^{i} − B_{jk}^{ir}E_{r},
E_{jkl}^{i} = ∂E_{jk}^{i}/∂x^{l} − (∂E_{jk}^{i}/∂y^{r})E_{lt}^{r}y^{t} − ∂E_{jl}^{i}/∂x^{k} + (∂E_{jl}^{i}/∂y^{r})E_{kt}^{r}y^{t}+
+ E_{jk}^{r}E_{rl}^{i} − E_{jl}^{r}E_{rk}^{i},
E_{jkl}^{′i} = G_{jkl}^{i} − B_{jkl}^{ir}E_{r}.

and the symbol ; means the h-covariant derivative with respect to the new Finsler connection (E_{jk}^{i}, E_{jk}^{i}y^{k}).

Finally, we deal with the v-curvature tensor S_{ijkl}(= C_{ilr}C_{j k}^{r} − C_{ikr}C_{j l}^{r}) of an n-dimensional Finsler space F. Iff U = L^{2}g^{il}g^{jk}S_{ijkl}, then the function U satisfies two conditions of Theorem 2.1. So, we have Theorem 3.3. Let F = (M, L) be an n-dimensional Finsler space with detU_{j}^{i} ≠ 0. A necessary and sufficient condition for F to be conformally flat is that

∂U_{i}/∂y^{j} = 0, U_{i;j} − U_{j;i} = 0, U_{jkl}^{i} = 0, U_{jkl}^{′i} = 0,

where

U = L^{2}g^{il}g^{jk}S_{ijkl}, U_{k}^{r} = (∂U/∂y^{i})B_{k}^{ir},
U_{j} = −E_{|k}D_{j}^{′k}, U_{j}^{′k} = (E_{j}^{k})^{−1}, U_{jk}^{i} = G_{jk}^{i} − B_{jk}^{ir}U_{r},
U_{jkl}^{i} = ∂U_{jk}^{i}/∂x^{l} − (∂U_{jk}^{i}/∂y^{r})U_{lt}^{r}y^{t} − ∂U_{jl}^{i}/∂x^{k} + (∂U_{jl}^{i}/∂y^{r})U_{kt}^{r}y^{t}+
+ U_{jk}^{r}U_{rl}^{i} − U_{jl}^{r}U_{rk}^{i},
U_{jkl}^{′i} = G_{jkl}^{i} − B_{jkl}^{ir}U_{r}.

and the symbol ; means the h-covariant derivative with respect to the new Finsler connection (E_{jk}^{i}, E_{jk}^{i}y^{k}).

References

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Department of Mathematics Faculty of Science Science University of Tokyo Sinjuku-ku, Tokyo, Japan