Landsberg Spaces Satisfying the T-Condition

Fumio Ikeda

Abstract

The T -tensor of Finsler spaces was studied by several authors ([1], [3], [4], [8], [10], [12] and [13]). Especially, Szabó [12], Watanabe and the present author [13] proved that a positive definite Finsler space with T - condition (i.e. T -tensor=0) is a Riemannian space. And non- Riemannian Finsler spaces with T -condition are treated by Asanov and Kirnasov [1].

The purpose of the present paper is to study non-Riemannian Landsberg spaces satisfying the T -condition and to consider conformal flatness of non-Riemannian Finsler spaces with T -condition. In §1, we shall define the T -tensor of Finsler spaces and show a Ricci identity and two Bianchi identities which play important roles. And, in §2, we shall prove that certain Landsberg spaces satisfying the T -condition become locally Minkowski spaces. Moreover, we shall show that Finsler spaces with T -condition are conformally flat if and only if they should be locally Minkowski spaces in §3. Finally, we shall apply above considerations to semi C-reducible Finsler spaces in §4.

Mathematics Subject Classification: 53C60 Key Words: Finsler spaces, Landsberg spaces, T -tensors.

1 Preliminaries

Let M^{n} be an n(≥ 3)-dimensional Finsler space endowed with a fundamental function L = L(x, y) and C_{ijk} be the (h)hv-torsion tensor of M^{n}, where x = (x^{i}) is a point and y = (y^{i}) is a supporting element of M^{n}, respectively. Then, the T -tensor T_{ijkl} of M^{n} is defined by the following relation

(1.1)  T_{ijkl} = C_{ijk} |_{l} +L^{−1}(C_{ijk}l_{l} + C_{ljk}l_{i} + C_{ilk}l_{j} + C_{ijl}l_{l}),

where l_{i} = L^{−1}g_{ir}y^{r}, g_{ir} = (∂L^{2}/∂y^{i}∂y^{r})/2 is called the fundamental tensor and the symbol | means the v-covariant derivative with respect to the Cartan connection CΓ of M^{n}. Definition 1.1. If the T -tensor T_{ijkl} of an n-dimensional Finsler space M^{n} vanishes identically, then M^{n} satisfies the T -condition.

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Transvecting (1.1) by the reciprocal tensor g^{ij} of g_{ij}, we obtain

(1.2)  T_{kl} = C_{k} |_{l} +L^{−1}(C_{k}l_{l} + C_{l}l_{k}),

where T_{kl} and C_{k} are called the T′-tensor and the torsion vector of M^{n}, respectively.

For the later use, we show a Ricci identity with respect to the torsion vector C_{i} and two Bianchi identities

(1.3)  C_{i|j|k} − C_{i|k|j} = −C_{r}R_{i jk}^{r} − C_{i} |_{r} R_{jk}^{r},

(1.4)  \begin{aligned}R_{jr}^{h}C_{k i}^{r} & − & R_{kr}^{h}C_{j i}^{r} + P _{jr}^{h}P _{ki}^{r} − P _{kr}^{h}P _{ji}^{r} + P _{ki|j}^{h}− \\ & − & P _{ji|k}^{h} + R_{jk}^{h} |_{i} −R_{i jk}^{h} = 0, \end{aligned}

(1.5)  \begin{aligned}R_{l jr}^{h}C_{k i}^{r} & − & R_{l kr}^{h}C_{j i}^{r} + P_{l jr}^{h}P _{ki}^{r} − P_{l kr}^{h}P _{ji}^{r} + P_{l ki|j}^{h}− \\ & − & P_{l ji|k}^{h} + R_{l jk}^{h} |_{i} +S_{l ir}^{h}R_{jk}^{r} = 0, \end{aligned}

where R_{i jk}^{r}, P_{i jk}^{r}, S_{i jk}^{r}, R_{jk}^{r} and P _{jk}^{r} are the h-curvature tensor, the hvcurvaure tensor, the v-curvature tensor, the (v)h-torsion tensor and the (v)hvtorsion tensor of M^{n}, respectively. And the symbol _{|} represents the h-covariant derivative with respect to CΓ.

2 Landsberg spaces

In this section, we shall deal with n(≥ 3)-dimensional Landsberg spaces defined by Definition 2.1. An n-dimensional Finsler space M^{n} is called a Landsberg space, if the (v)hv-torsion tensor P _{jk}^{r} of M^{n} vanishes identically.

Let M^{n} be an n(≥ 3)-dimensional Landsberg space with T -condition, then (1.2), (1.4) and (1.5) reduce to

(2.1)  C_{k} |_{l}= −L^{−1}(C_{k}l_{l} + C_{l}l_{k}),

(2.2)  R_{jr}^{h}C_{k i}^{r} − R_{kr}^{h}C_{j i}^{r} + R_{jk}^{h} |_{i} −R_{i jk}^{h} = 0,

(2.3)  R_{l jr}^{h}C_{k i}^{r} − R_{l kr}^{h}C_{j i}^{r} + R_{l jk}^{h} |_{i} +S_{l ir}^{h}R_{jk}^{r} = 0,

because the hv-curvature tensor P_{i jk}^{r}, the (v)hv-torsion tensor P _{jk}^{r} and the T′-tensor T_{kl} vanish identically.

It is well-known that Landsberg spaces satisfying the T -condition become Berwald spaces ([4] and [8]), then the h-covariant derivative of the (h)hv-torsion tensor C_{i k}^{r} vanishes identically, and so C_{i|j} = 0 holds. Thus, (1.3) and (2.1) yield

(2.4)  −C_{r}R_{i jk}^{r} + L^{−1}C_{r}R_{jk}^{r}l_{i} = 0,

because of l_{r}R_{jk}^{r} = 0.

V -covariantly differentiating (2.4) and using (2.1), we find L^{−1}C_{r}R_{i jk}^{r}l_{l} − L^{−2}R_{ijk}C_{l} − C_{r}R_{i jk}^{r} |_{l} −2L^{−2}C_{r}R_{jk}^{r}l_{i}l_{l} + L^{−1}C_{r}R_{jk}^{r} |_{l} l_{i} + L^{−2}C_{r}R_{jk}^{r}h_{il} = 0, which leads, by virtue of (2.2) and (2.3), to

(2.5)  L^{−2}C_{r}R_{jk}^{r}h_{il} − L^{−2}R_{ijk}C_{l} + C_{r}S_{i lt}^{r}R_{jk}^{t} = 0.

where h_{il}(= g_{il} − l_{i}l_{l}) is the angular metric tensor of M^{n}.

Now we consider Landsberg spaces whose v-curvature tensor S_{ijkl} has a certain form. First, suppose that the v-curvature tensor S_{ijkl} of our Landsberg spaces vanishes identically. Then (2.5) reduces to

(2.6)  L^{−2}C_{r}R_{jk}^{r}h_{il} − L^{−2}R_{ijk}C_{l} = 0.

Transvection of (2.6) by g^{il} yields (n − 2)L^{−2}C_{r}R_{jk}^{r} = 0, from which, via (2.2) and (2.6), R_{i jk}^{r} = 0 is derived. Therefore, we have Theorem 2.1. If v-curvature tensor S_{i jk}^{r} of an n(≥ 3)-dimensional Landsberg space M^{n} with T -condition vanishes identically, then M^{n} is a locally Minkowski space.

Since the v-curvature tensor S_{ijkl} of a C2-like Landsberg space M^{n} vanishes identically, it is evident the following corollary: Corollary 2.2. If an n(≥ 3)-dimensional C2-like Landsberg space M^{n} satisfies the T -condition, then M^{n} is a locally Minkowski space.

Next, we assume that our Landsberg spaces are S3-like, and then the vcurvature tensor S_{ijkl} is represented by

(2.7)  S_{ijkl} = S(h_{ik}h_{jl} − h_{il}h_{jk}).

Substituting (2.7) into (2.5) and then transvecting it by g^{il}, we have

(2.8)  (L^{−2} + S)(C_{r}R_{jk}^{r}h_{il} − R_{ijk}C_{l}) = 0.

Thus, if L^{−2} ≠ −S holds, then (2.2) and (2.8) lead to R_{i kl}^{j} = 0. Therefore, we get Theorem 2.3. Let M^{n} be an n(≥ 3)-dimensional S3-like Landsberg space M^{n} with T -condition. If the function S of (2.7) is not equal to −L^{−2}, then M^{n} is a locally Minkowski space.

Since the v-curvature tensor S_{ijkl} of three-dimensional Finsler spaces is written as (2.7), the following corollary is evident: Corollary 2.4. If a three-dimensional Landsberg space M^{3} satisfies the T - condition and the function S of (2.7) is not equal to −L^{−2}, then M^{3} is a locally Minkowski space.

3 Conformal flatness of Finsler spaces with T -condition

In this section, we shall deal with conformal flatness of Finsler spaces. For this concept, Kikuchi proved a very important theorem under the Kikuchi condition that the tensor W _{j}^{i} is regular, where W _{j}^{i} = (∂L^{2}C^{2}/∂y^{r})B_{j}^{ri}, B_{j}^{ri} = ∂B^{ri}/∂y^{j} and B^{ri} = L^{2}(g^{rj} − l^{r}l^{i})/2 [8]. If Finsler spaces satisfy the T -condition, then the above tensor W _{j}^{i} is equal to zero. So, Finsler spaces with T -condition don’t satisfy the Kikuchi condition.

We have already known that if an n(≥ 3)-dimensional Finsler space M^{n} satisfies the T -condition, then the (v)hv-torsion tensor P_{ijk} of M^{n} is conformally invariant [2]. Thus, it is clear that an n(≥ 3)-dimensional Finsler space M^{n} satisfying the T -condition is conformal to a Landsberg space if and only if M^{n} is a Landsberg space. Therefore, from above results, Theorem 2.1 and Theorem 2.3, we have the following theorems: Theorem 3.1. If the v-curvature tensor S_{i jk}^{r} of an n(≥ 3)-dimensional Finsler space M^{n} with T -condition vanishes identically, then a necessary and sufficient condition for M^{n} to be conformally flat is that M^{n} itself is a locally Minkowski space. Theorem 3.2. If an n(≥ 3)-dimensional Finsler space M^{n} with T -condition is S3-like and the function S of (2.7) is not equal to −L^{−2}, then M^{n} is conformally flat if and only if M^{n} itself is a locally Minkowski space. Theorem 3.3. If a three-dimensional Landsberg space M^{3} is the T -condition and the function S of (2.7) is not equal to −L^{−2}, then M^{3} is conformally flat if and only if M^{n} itself is a locally Minkowski space.

4 Semi C-reducible Finsler spaces

In this section, we shall consider semi C-reducible Finsler spaces whose (h)hvtorsion tensor C_{ijk} is written as

(4.1)  C_{ijk} = p(h_{ij}C_{k} + h_{jk}C_{i} + h_{ki}C_{j}) + qC_{i}C_{j}C_{k},

where p, q are functions satisfying (n + 1)p + qC^{2} = 1 and C^{2} = g_{ij}C^{i}C^{j}.

From the definition of the v-curvature tensor S_{ijkl}(= C_{ilt}C_{j k}^{t} −C_{ikt}C_{j l}^{t}) and the equation (4.1), it is derived

(4.2) \begin{aligned}S_{ijkl} & = & p^{2}C^{2}(h_{il}h_{jk} − h_{ik}h_{jl})+ \\ & + & (p^{2} + pqC^{2})(h_{il}C_{j}C_{k} + h_{jk}C_{i}C_{l} − h_{ik}C_{j}C_{l} − h_{jl}C_{i}C_{k})). \end{aligned}

Substituting (4.2) into (2.5) and then transvecting it by g^{il}, we have

(4.3)  (n − 2)(L^{−2} − pC^{2} + (n − 1)p^{2}C^{2})(C_{r}R_{jk}^{r}h_{il} − R_{ijk}C_{l}) = 0.

Therefore, we obtain Theorem 4.1. Let M^{n} be an n(≥ 3)-dimensional semi C-reducible Landsberg space. If M^{n} satisfies the T -condition and L^{−2} − pC^{2} + (n − 1)p^{2}C^{2} ≠ 0, then M^{n} is a locally Minkowski space. Theorem 4.2. Let M^{n} be an n(≥ 3)-dimensional semi C-reducible Finsler space with T -condition. If M^{n} satisfies L^{−2} − pC^{2} + (n − 1)p^{2}C^{2} ≠ 0, then M^{n} is conformally flat if and only if M^{n} itself is a locally Minkowski space.

Since the function p of C-reducible Finsler spaces is (n + 1)^{−1}, the following theorems are evident.

Theorem 4.3. Let M^{n} be an n(≥ 3)-dimensional C- reducible Landsberg space. If M^{n} satisfies the T -condition and L^{2}C^{2} ≠ (n + 1)^{2}/2, then M^{n} is a locally Minkowski space. Theorem 4.4. Let M^{n} be an n(≥ 3)-dimensional C-reducible Finsler space with T -condition. If M^{n} satisfies L^{2}C^{2} ≠ (n + 1)^{2}/2, then M^{n} is conformally flat if and only if M^{n} itself is a locally Minkowski space.

Next, we shall state special semi C-reducible Finsler spaces whose (h)hvtorsion tensor represented by

(4.4)  C_{ijk} = (h_{ij}C_{k} + h_{jk}C_{i} + h_{ki}C_{j})/(n − 1) − 2C^{−2}C_{i}C_{j}C_{k}/(n − 1).

Such semi C-reducible Finsler spaces were treated by Izumi, Sakaguchi and Yoshida in their paper [7]. The function L^{−2} − pC^{2} + (n − 1)p^{2}C^{2} is equal to L^{−2}, because of p = (n − 1)^{−1}. Therefore, from Theorem 4.1 and 4.2, we have Theorem 4.5. Let M^{n} be an n(≥ 3)-dimensional Landsberg space whose (h)hvtorsion tensor C_{ijk} is represented by (4.4). If M^{n} satisfies the T -condition, then M^{n} is a locally Minkowski space. Theorem 4.6. Let M^{n} be an n(≥ 3)-dimensional Finsler space with T -condition. If the (h)hv-torsion tensor C_{ijk} is presented by (4.4) then a necessary and sufficient condition for M^{n} to be conformally flat is that M^{n} itself is a locally Minkowski space.

Finally, we shall consider three-dimensional Finsler spaces whose (h)hv- torsion tensor C_{ijk} is represented by (4.4). Let (l_{i}, m_{i}, n_{i}) be a Moór frame and H, I and J be main scalas, then (4.4) rewritten as

(4.5)  LC_{ijk} = I(m_{i}m_{j}m_{k} + m_{i}n_{j}n_{k} + n_{i}m_{j}n_{k} + n_{i}n_{j}m_{k}),

which shows that H = I and J = 0 hold. From the previous paper [5], I |_{t} n^{t} = 0 is satisfied because of J = 0, and then the T -tensor T_{ijkl} is rewritten as (4.6)

\begin{aligned}T_{ijkl} & = & I |_{t} m^{t}(m_{i}m_{j}m_{k}m_{l} − 3n_{i}n_{j}n_{k}n_{l} + m_{i}m_{j}n_{k}n_{l} + m_{i}n_{j}m_{k}n_{l}+ \\ & + & m_{i}n_{j}n_{k}m_{l} + n_{i}n_{j}m_{k}m_{l} + n_{i}m_{j}n_{k}m_{l} + n_{i}m_{j}m_{k}n_{l}), \end{aligned}

and the V -curvature tensor S_{ijkl} vanishes identically.

On the other hand, the Kikuchi condition for conformal flatness of Finsler spaces becomes II |_{i}≠ 0, which is equivarent to T_{ijkl} ≠ 0. Summarizing these results, we have Theorem 4.7 Let M^{3} be a three-dimensional Finsler spaces whose (h)hv-torsion tensor C_{ijk} is written as (4.4). If M^{3} does not satisfy the Kikuchi condition for conformal flatness, then a necessary and sufficient condition for M^{3} to be conformally flat is that M^{3} itself is a locally Minkowski space.

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