Finsler Spaces Admitting a Parallel Vector Field

Masashi Kitayama

Abstract In this paper we modify the fundamental function of a Finsler space with the help of a parallel Finsler vector field, getting a new Finsler space whose properties are investigated.

Mathematics Subject Classification: 53C60 Key Words: modified Finsler metric, particular Finsler spaces.

The terminology and notations are referred to Matsumoto’s monograph [5]. Let F^{n} be an n–dimensional Finsler space with a fundamental function

L(x, y) (y = ẋ), and we shall introduce in F^{n} the Cartan connection CΓ = (F_{j}^{i}_{k}, N^{i}_{k}, C_{j}^{i}_{k}).

Let us consider a vector field X^{i}(x) in F^{n}: this field is called parallel, if it satisfies the partial differential equations

\begin{matrix}(1) & X^{i}_{|j} := ∂_{j}X^{i} − N^{h}_{j} ˙∂_{h}X^{i} + X^{h}F_{h}^{i}_{j} = ∂_{j}X^{i} + X^{h}F_{h}^{i}_{j} = 0, \\ (2) & X^{i}|_{j} := ˙∂_{j}X^{i} + X^{h}C_{h}^{i}_{j} = X^{h}C_{h}^{i}_{j} = 0, \end{matrix}

where ∂_{j} and ˙∂_{j} denote partial differentiations by x^{j} and y^{j}, respectively. From the Ricci identities, the following integrability conditions hold:

\begin{matrix}(3) & X^{h}R_{hijk} = 0, \\ (4) & X^{h}P_{hijk} = 0, \\ (5) & X^{h}S_{hijk} = 0, \end{matrix}

where R_{hijk}, P_{hijk} and S_{hijk} are the components of the curvature tensors of CΓ.

Remark. In particular, the above equations (1) and (2) are satisfied for a stationary vector field X^{i}(x).

In terms of covariant components X_{i}(x)’s, (1) and (2) are written as

\begin{matrix}(1)′ & X_{i|j} = 0, \\ (2)′ & X_{i}|_{j} = 0. \end{matrix}

Balkan Journal of Geometry and Its Applications, Vol.3, No.2, 1998, pp. 29-36 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Here we shall consider the modification of a Finsler metric by a parallel vector field X^{i}(x), as follows. Putting

(6)  ^{∗}L^{2} = L^{2} + β^{2} (β = X_{i}(x)y^{i} ≠ 0),

∗L defines a new Finsler metric of M. It is said that ^{∗}L is obtained by a β– change of the metric L [6]. The metric tensor derived from ^{∗}L is written as follows:

(7)  ^{∗}g_{ij} = g_{ij} + X_{i}X_{j}, ^{∗}g^{ij} = g^{ij} − X^{i}X^{j}/(1 + X^{2}),

where X is the length of X^{i} with respect to the original metric. The coefficients of Cartan’s connection are written in the forms

(8)  ^{∗}N^{i}_{j} = N^{i}_{j}, ^{∗}F_{j}^{i}_{k} = F_{j}^{i}_{k},

and we have ^{∗}F_{ijk} = (δ^{h}_{j} + X_{j}X^{h})F_{ihk}.

From (7) we immediately get

(9)  ^{∗}C_{ijk} = C_{ijk}, ^{∗}C_{j}^{i}_{k} = C_{j}^{i}_{k}.

As a consequence, (8) and (9) yield

Proposition 1. If a Finsler space with a fundamental function L admits a parallel vector field X^{i}, then the vector field X^{i} is parallel with respect to the modified metric (6).

In view of (9), we can conclude immediately

(10)  ^{∗}S_{hijk} = S_{hijk}.

The components of the other two curvature tensors are

\begin{matrix}(11) & ^{∗}R_{hijk} = R_{hijk}, & ^{∗}R_{h}^{i}_{jk} = R_{h}^{i}_{jk}, \\ (12) & ^{∗}P_{hijk} = P_{hijk}, & ^{∗}P_{h}^{i}_{jk} = P_{h}^{i}_{jk}, \end{matrix}

For the later use, we shall show here two lemmas.

Lemma 1. ([6]) The covariant vector m_{i}(:= X_{i} −βy_{i}/L^{2}) is a non–zero vector orthogonal to y^{i}. Proof. Assuming that m_{i} = 0, we have L^{2}X_{i} − βy_{i} = 0. Differentiating this

by y^{j} and denoting ˙∂_{i}L(= y_{i}/L) by ℓ_{i}, we are led to a contradiction h_{ij}(:= g_{ij} − ℓ_{i}ℓ_{j}) = 0.

Lemma 2. For the angular metric tensor h_{ij} and the covariant vector m_{i}, we have

\begin{matrix}(13) & h_{ij}X^{j} = m_{i} & (≠ 0), \\ (14) & m_{i}X^{i} = m^{2} & (≠ 0), \end{matrix}

where m^{2} = g_{ij}m^{i}m^{j} and m^{i} = g^{ij}m_{j}.

Proof. Assuming that m^{2} = 0, we get L^{2}X^{2}−β^{2} = 0. Then ˙∂_{j} ˙∂_{i}(L^{2}X^{2}−β^{2}) = 0 gives us g_{ij} = X_{i}X_{j}/X^{2},which contradicts rank(g_{ij}) = n.

Remark. It follows from β ≠ 0 that the covariant vectors m_{i} and X_{i} are non–zero vectors.

Now, we shall consider the T –condition

(15)  T_{hijk} := LC_{hij}|_{k} + ℓ_{h}C_{ijk} + ℓ_{i}C_{hjk} + ℓ_{j}C_{hik} + ℓ_{k}C_{hij} = 0,

where the T –tensor T_{hijk} is completely symmetric.

If we contract (15) by X^{h}, we have X^{h}ℓ_{h}C_{ijk} = 0 in virtue of (2). But from X^{h}ℓ_{h} = β/L ≠ 0, we find C_{ijk} = 0. Consequently, from (9) we find

Theorem 1. If a Finsler space F^{n} satisfying (15) admits a parallel vector field, then both the Finsler spaces F^{n} and ^{∗}F^{n} are Riemannian.

The generalized T –condition is defined by

(16)  T_{ij} := T_{ijrs}g^{rs} = LC_{i}|_{j} + ℓ_{i}C_{j} + ℓ_{j}C_{i} = 0,

where the tensor T_{ij} is called the contracted T –tensor and C_{i} = C_{ijk}g^{jk} being the torsion vector.

Contracting (16) by X^{i} and using (2), we have C_{j} = 0. According to Deicke’s theorem and (9), we have

Theorem 2. If a Finsler space F^{n} satisfying (16) admits a parallel vector field, then both the Finsler spaces F^{n} and ^{∗}F^{n} are Riemannian.

We are concerned with a space of scalar curvature in Berwald’s sense. It is characterized by the equation

(17)  R_{i0k}(= R_{ijk}y^{j}) = L^{2}Kh_{ik},

or

(18)  R_{ijk} = h_{ik}K_{j} − h_{ij}K_{k}, K_{j} = L^{2} ˙∂_{j}K/3 + LKℓ_{j},

where h_{ik} is the angular metric tensor and the scalar curvature K is a Finsler scalar field.

Contracting (17) by X^{i} and using (3) and (13), we obtain

Proposition 2. If a Finsler space F^{n} of scalar curvature K admits a parallel vector field, then the scalar curvature K vanishes.

From Proposition 2 and (18), we immediately get R_{ijk} = 0. In terms of (9) and (11), the Berwald’s curvature tensor H_{hijk}(:= ˙∂_{h}R_{ijk} − 2C_{ihr}R^{r}_{jk}) coincides with ^{∗}H_{hijk}, so we find

Theorem 3. If a Finsler space F^{n} of scalar curvature K admits a parallel vector field, then both Berwald’s curvature tensors H_{hijk} and ^{∗}H_{hijk} vanish.

A Finsler space F^{n} (n > 2) is called quasi–C–reducible, if the torsion tensor C_{ijk} is written as

(19)  C_{ijk} = A_{ij}C_{k} + A_{jk}C_{i} + A_{ki}C_{j},

where A_{ij} is a symmetric Finsler tensor field satisfying A_{i0}(:= A_{ij}y^{j}) = 0.

Contracting (19) by X^{i}X^{j} and using (2), we immediately get λC_{k} = 0, where λ = A_{ij}X^{i}X^{j}. Therefore, taking into account (9) we have Proposition 3. If a quasi–C–reducible Finsler space F^{n} (n > 2) admits a parallel vector field, then both the Finsler spaces F^{n} and ^{∗}F^{n} are Riemannian, provided that λ(= A_{ij}X^{i}X^{j}) ≠ 0.

The condition we consider next is the C–reducibility. A Finsler space F^{n} (n > 2) which satisfies the equation

(20)  (n + 1)C_{ijk} = h_{ij}C_{k} + h_{jk}C_{i} + h_{ki}C_{j},

is said to be C–reducible.

Contracting (20) by X^{i}X^{j} and using (2), we obtain h_{ij}X^{i}X^{j}C_{k} = 0. Paying attention to Lemma 2, we get C_{k} = 0. Thus, taking into account of (9) we have

Theorem 4. If a C–reducible Finsler space F^{n} (n > 2) admits a parallel vector field, then both the Finsler spaces F^{n} and ^{∗}F^{n} are Riemannian.

A Finsler space F^{n} (n > 2) with non–zero length C of the torsion vector C^{i} is called semi–C–reducible, if the torsion tensor C_{ijk} is of the form

(21)  C_{ijk} = p(h_{ij}C_{k} + h_{jk}C_{i} + h_{ki}C_{j})/(n + 1) + qC_{i}C_{j}C_{k}/C^{2},

where C^{2} = g^{ij}C_{i}C_{j} and p + q = 1.

As the exceptional case (p = 0) of semi–C–reducibility, we are led to the following definition:

A Finsler space F^{n} (n ≥ 2) with C^{2} ≠ 0 is called C2–like, if the torsion tensor C_{ijk} is written in the form

(22)  C_{ijk} = C_{i}C_{j}C_{k}/C^{2}.

Contracting (21) by X^{i}X^{j} and using (2) and Lemma 2, we get pC_{k} = 0. Then we obtain p = 0 because of C^{2} ≠ 0. In virtue of p + q = 1, we have q = 1. Consequently, taking into account of (9), we obtain

Theorem 5. If a semi–C–reducible Finsler space F^{n} (n > 2) admits a parallel vector field, then both the Finsler spaces F^{n} and ^{∗}F^{n} are C2–like.

A Finsler space F^{n} (n ≥ 2) will be called C^{h}–recurrent, if the torsion tensor C_{ijk} satisfies the equation

(23)  C_{ijk|ℓ} = C_{ijk}K_{l},

where K_{ℓ} = K_{ℓ}(x, y) is a covariant vector field.

The following expressions are well–known,

\begin{matrix}(24) & P_{hijk} = U_{(hi)}\{C_{ijk|h} + C_{hjr}C_{i}^{r}_{k|0}\}, \\ (25) & P_{ijk} = C_{ijk|0}, \\ (26) & S_{hijk} = U_{(jk)}\{C_{hkr}C_{i}^{r}_{j}\}, \end{matrix}

where the index 0 means contraction by y^{i} and the notation U_{(hi)} denotes the interchange of indices h, i and subtraction.

Contracting (24) by X^{h} and using (1),(2) and (4), we have

Lemma 3. For a torsion tensor C_{ijk} and a parallel vector field X^{h}, we have

(27)  X^{h}C_{ijk|h} = 0.

Contracting (23) by X^{ℓ} and using (27) and (9), we obtain

Proposition 4. If a C^{h}–recurrent Finsler space F^{n} (n > 2) admits a parallel vector field, then both the Finsler spaces F^{n} and ^{∗}F^{n} are Riemannian, provided that μ(= X^{ℓ}K_{ℓ}) ≠ 0.

Diferentiating (26) h-covariantly we obtain

(28)  S_{hijk|ℓ} = U_{(jk)}{C_{hkr|ℓ}C_{i}^{r}_{j} + C_{hkr}C_{ij|ℓ}^{r}}.

Contracting this by X^{ℓ} and using (27), we get X^{ℓ}S_{hijk|ℓ} = 0.

A P 2–like Finsler space F^{n} (n > 2) is characterized by

(29)  P_{hijk} = K_{h}C_{ijk} − K_{i}C_{hjk},

where K_{h} = K_{h}(x, y) is a covariant vector field.

Contracting this by X^{h} and using (2) and (4) we get X^{h}K_{h}C_{ijk} = 0. Therefore, taking into acount (9) we have Theorem 6. If a P2-like Finsler space F^{n} (n > 2) admits a parallel vector field, then both the Finsler spaces F^{n} and ∗ F^{n} are Riemannian providing that ν(= X^{h}K_{h}) ≠ 0.

A Landsberg space is characterized by P_{ijk}(= C_{ijk|0}) = 0. Further, a Finsler space is called P –reducible, if the torsion tensor P_{ijk} is written as

(30)  P_{ijk} = (h_{ij}P_{k} + h_{jk}P_{i} + h_{ki}P_{j})/(n + 1),

where P_{i} = P^{r}_{ir} = C_{i|0}.

Contracting (30) by X^{i}X^{j} and using (25),(1) and (2), we obtain X^{i}X^{j}h_{ij}P_{k} = 0. From Lemma 2, we get P_{k} = 0. Thus, taking into account (12), we have

Theorem 7. If a P –reducible Finsler space F^{n} admits a parallel vector field, then both the Finsler spaces F^{n} and ^{∗}F^{n} are Landsberg spaces.

A Finsler space F^{n} (n > 3) is called S3–like if the curvature tensor S_{hijk} is written in the form

(31)  L^{2}S_{hijk} = S(h_{hj}h_{ik} − h_{hk}h_{ij}),

where the scalar curvature S(= S_{hijk}g^{hj}g^{ik}) is a function of position alone.

Contracting (31) by X^{h}g^{ik} and using (5), (13) and Lemma 1, we get S = 0. Therefore, taking into account (10), we have

Theorem 8. If an S3–like Finsler space F^{n} (n > 3) admits a parallel vector field, then both the curvature tensors S_{hijk} and ^{∗}S_{hijk} vanish.

Similar to the case of the S3–likeness, we are concerned with the following:

A Finsler space F^{n} (n > 4) is called S4–like if the curvature tensor S_{hijk} is written in the form

(32)  L^{2}S_{hijk} = h_{hj}M_{ik} + h_{ik}M_{hj} − h_{hk}M_{ij} − h_{ij}M_{hk},

where M_{ij} is a symmetric and indicatory tensor. Then the tensor M_{ij} of the above definition is given by

(33)  M_{hi} = L^{2}{S_{hi} − Sh_{hi}/2(n − 2)}/(n − 3),

where S_{hi} = S_{h}^{r}_{ir}.

Here we shall prove a lemma.

Lemma 4. For the angular metric tensors we have

\begin{matrix}(34) & ^{∗}h_{ij} = h_{ij} + τm_{i}m_{j} (τ = L^{2}/^{∗}L^{2}), \\ (35) & h_{ij} = M_{ij}/ν + 2m_{i}m_{j}/m^{2} (ν = L^{2}S/2(n − 2)(n − 3)). \end{matrix}

Proof. Contracting (32) and (33) by X^{h} and using (5) and Lemma 2, we obtain ν(h_{ik}m_{j} −h_{ij}m_{k}) = M_{ik}m_{j} −M_{ij}m_{k}. Further contracting this by X^{j} and using Lemma 2, we get (35).

From (5), (7), (10), (33) and (34), we have

∗S_{hi} = S_{hi}, ^{∗}S = S, ^{∗}M_{hi} = M_{hi}/τ − νm_{h}m_{i}.

Thus, using (10) and Lemma 4, we obtain

Theorem 9. If an S4–like Finsler space (n > 4) F^{n} admits a parallel vector field X^{i}, then the modified metric (6) is also S4–like.

A Finsler space F^{n} (n > 2) is said to be of h–isotropic, if the curvature tensor R_{hijk} is written as

(36)  R_{hijk} = K(g_{hj}g_{ik} − g_{hk}g_{ij}),

where K is a Finsler scalar. In 1961 Akbar–Zadeh proves that K is a constant.

Contracting (36) by X^{h}m^{j} and using (3) and (14), we obtain K(m^{2}g_{ik} − m_{i}X_{k}) = 0. Further, contracting this by g^{ik}, we have K = 0. Hence, taking into account (11), we get

Theorem 10. If an h–isotropic Finsler space F^{n} (n > 2) admits a parallel vector field, then both the curvature tensors R_{hijk} and ^{∗}R_{hijk} vanish.

Next, we shall be concerned with the notion of an R3–like Finsler space which is defined by the following:

A Finsler space F^{n} (n > 3) is called R3–like, if the curvature tensor R_{hijk} is written in the form

(37)  R_{hijk} = g_{hj}L_{ik} + g_{ik}L_{hj} − g_{hk}L_{ij} − g_{ij}L_{hk}.

It follows from (37) that R_{ik} = g^{hj}R_{hijk} = (n − 2)L_{ik} + Lg_{ik}, where L = L_{ij}g^{ij} and R = g^{ik}R_{ik} = 2(n − 1)L. Thus we obtain L_{ik} depending on R_{ik} and R. Since R_{ik} are not symmetric in general, so are L_{ik}. Further, we have to calculate ∗R_{ik} and ^{∗}R. Using (3),(7) and (11) the results are as follows:

∗R_{ik} = R_{ik}, ^{∗}R = R.

Hence, substituting these in the formula giving L_{ik}, we obtain

∗L_{ik} = L_{ik} − RX_{i}X_{k}/2(n − 1)(n − 2).

Using this and (7), and taking into account (3), (11) and (37), we have

Theorem 11. If an R3–like Finsler space F^{n} (n > 3) admits a parallel vector field X^{i}, then the modified metric (6) is also R3–like.

References

[1] M.Matsumoto: On h–isotropic and C^{h}–recurrent Finsler spaces, J. Math. Kyoto Univ. 11–1(1971), 1–9.

[2] M.Matsumoto and K.Eguchi: Finsler spaces admitting a concurrent vector field, Tensor, N.S. Vol. 28(1974), 239–249.

[3] M.Matsumoto: On Finsler spaces with Randers’ metric and special forms of important tensors, J. Math. Kyoto Univ. 14–3(1974), 477–498.

[4] M.Matsumoto and C.Shibata: On semi–C–reducibility, T –tensor=0 and S4–likeness of Finsler spaces, J. Math. Kyoto Univ. 19–2(1979), 301–314.

[5] M.Matsumoto: Foundations of Finsler Geometry and Special Finsler Spaces, Kaiseisha Press, Saikawa, Ōtsu, Japan, 1986.

[6] C.Shibata: On invariant tensors of β–changes of Finsler metrics, J. Math. Kyoto Univ. 24–1(1984), 163–188.

Department of Mathematics Kushiro Campus, Hokkaido University of Education Kushiro, Hokkaido 085 JAPAN