Conformally Closed Finsler Spaces

Makoto Matsumoto

Abstract

Let S be a set of a special kind of Finsler spaces. If F^{n} ∈ S remains to belong to S by any conformal change of metric, then S is called conformally closed. The present paper is devoted mainly to studying conformally closed sets of Berwald spaces and Douglas spaces.

Mathematics Subject Classification (1991): 53B40 Key words: Finsler metrics, conformal change of metrics.

1 Introduction

Let us denote by F^{n} = (M^{n}, L(x, y)) an n-dimensional Finsler space on a smooth n-manifold M^{n} with the fundamental metric function L(x, y), x = (x^{i}), y = (y^{i}). In the present paper we are concerned with the theory of conformal changes of metrics:

(1.1)  F^{n} = (M^{n}, L(x, y)) → F̄^{n} = (M^{n}, L̄(x, y)), L̄ = e^{c(x)}L,

with the conformal factor c(x).

M. Hashiguchi ([5], 1976) has developed the theory based on the new formulation of Finsler connections initiated by the present author [9]. In the first place he found conformally invariant tensors

(1.2)  B_{ij} = (g_{ij} − 2l_{i}l_{j})/F, B^{ij} = F (g^{ij} − 2l^{i}l^{j}),

where F = L^{2}/2 and (B^{ij}) is the inverse of the matrix (B_{ij}).

Secondly he dealt with the quantities G^{i}(x, y), from which the Berwald connection BΓ = (G_{j k}^{i}, G_{j}^{i}) is constructed; G_{j}^{i} = ˙∂_{j}G^{i} and G_{j k}^{i} = ˙∂_{k}G_{j}^{i}. On the conformal change (1.1) he showed

(1.3)  Ḡ^{i} = G^{i} − B^{ir}c_{r}, c_{r} = ∂_{r}c(x).

Then the changed Berwald connection (Ḡ_{j k}^{i}, Ḡ_{j}^{i}) of F̄^{n} is given as

(1.4)  Ḡ_{j}^{i} = G_{j}^{i} − B_{j}^{ir}c_{r}, Ḡ_{j k}^{i} = G_{j k}^{i} − B_{jk}^{ir}c_{r},

Balkan Journal of Geometry and Its Applications, Vol.4, No.1, 1999, pp. 117-128 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

where B_{j}^{ir} = ˙∂_{j}B^{ir} and B_{jk}^{ir} = ˙∂_{k}B_{j}^{ir}.

Thirdly he constructed the change of the Cartan connection CΓ = (F_{j k}^{i}, G_{j}^{i}, C_{j k}^{i}). Let us denote by (_{|i}, |i) the h- and v-covariant differentiations in CΓ. Then he showed

F̄_{j k}^{i} = F_{j k}^{i} − U_{jk}^{ir}c_{r}.

We treat the conformal invariants U_{jk}^{ir}. Putting V _{jk}^{ir} = U_{jk}^{ir} − B_{jk}^{ir}, we have

(1.5)  g_{hr}g_{is}V _{jk}^{rs} = LT_{hijk},

where the well-known T -tensor appears ([9]; [1], (3.5.3.1)), defined by

T_{hijk} = LC_{hij}|_{k} + (l_{h}C_{ijk} + _{4} ).

Here and throughout the following we shall use, as in [14], the abbreviations to avoid long expressions of the similar terms:

l_{h}C_{ijk} + _{4} = l_{h}C_{ijk} + l_{i}C_{hjk} + l_{j}C_{hik} + l_{k}C_{hij},

h_{hi}F_{jk} + _{6} = h_{hi}F_{jk} + h_{hj}F_{ik} + h_{hk}F_{ij} + h_{ij}F_{hk} + h_{ik}F_{hj} + h_{jk}F_{hi}.

Then he obtained the relation between the (v)hv-torsion tensors P _{jk}^{i} = C_{j k|0}^{i} as

(1.6)  P̄ _{jk}^{i} = P _{jk}^{i} + V _{jk}^{ir}c_{r}.

A Finsler space is called a Landsberg space ([1], [9]) if P _{jk}^{i} vanishes identically. As a consequence of (1.6) with (1.5) we obtain Hashiguchi’s theorem: A Landsberg space remains to be a Landsberg space by any conformal change of metric, if and only if its T -tensor vanishes identically. We should like to define the notion of ”conformally closed” by expressing this theorem as Theorem H. A Landsberg space is conformally closed, if and only if its T -tensor vanishes identically.

Since (1.5) shows T̄_{hijk} = e^{4c}T_{hijk}, the condition T = 0 is conformally invariant. Hence, if we define the two sets:

- L(n) ⋯ Landsberg spaces of dim . n,

- L_{c}(n) ⋯ conformally closed Landsberg spaces of dim . n, then any F^{n} ∈ L_{c}(n) remains to belong to L_{c}(n) by any conformal change of metric, while for any F^{n} ∈ L(n)\L_{c}(n) we have a function c(x) such that F̄^{n} = (M^{n}, e^{c}L) ∉ L(n). Thus L_{c}(n) may be said to be closed in L(n) with respect to conformal changes of metrics.

Further we should like to use ”conformally closed” in a sense as follows: Let us consider an F^{n} with the 1-form metric L(a^{α}) where a^{α} = a_{i}^{α}(x)y^{i} are n independent 1-forms of y^{i} [10]. Since L(a^{α}) is assumed to be positively homogeneous in a^{α} of degree one, the conformal change (1.1) yields L̄ = e^{c}L(a^{α}) = L(e^{c}a^{α}). Consequently L̄ = L(ā^{α}) is still a 1-form metric with ā^{α} = e^{c}a^{α}. This property may be said as follows: Proposition 1. The notion of the 1-form metric is conformally closed.

As a consequence we may say that the notion of the Riemannian metric is conformally closed and any Riemannian space is conformally closed. It is also obvious that any conformally flat Finsler space is conformally closed, because if F^{n} is conformal to a locally Minkowski space, then the conformally changed F̄^{n} of F^{n} is also conformal to the locally Minkowski space.

2 Conformally closed Berwald spaces, I

In Proposition 4 of the paper [13] the present author showed interesting Berwald spaces of dimension three: Example 1. Berwald spaces which are respectively conformal to Minkowski spaces with the cubic metrics L_{1} and L_{2}:

(L_{1})^{3} = ẋ^{3} + ẏ^{3} + ż^{3} − 3ẋẏż, (L_{2})^{3} = ẋẏż.

These Minkowski spaces (R^{3}, L_{1}) and (R^{3}, L_{2}) have the remarkable property: Any conformally changed spaces of these spaces are Berwald spaces. Definition. A Berwald space is called conformally closed, if it remains to be a Berwald space by any conformal change of metric. We denote by B_{c}(n) the set of all conformally closed Berwald spaces of dimension n.

Consequently the Minkowski spaces (R^{3}, L_{1}) and (R^{3}, L_{2}) in Example 1 belong to B_{c}(3).

Now (1.4) leads to the conformal change of the hv-curvature tensor G_{j kl}^{i} = ˙∂_{l}G_{j k}^{i} of BΓ:

(2.1)  Ḡ_{j kl}^{i} = G_{j kl}^{i} − B_{jkl}^{ir}c_{r}, B_{jkl}^{ir} = ˙∂_{l}B_{jk}^{ir}.

A Finsler space F^{n} is called a Berwald space ([1], [9]), if BΓ is linear, that is, G_{j k}^{i} are functions of position alone. Therefore F^{n} is a Berwald space, if and only if G_{j kl}^{i} vanishes identically. Hence (2.1) shows Proposition 2. A Berwald space is conformally closed, if and only if B_{jkl}^{ir} vanishes identically.

Since B_{jkl}^{ir} = ˙∂_{j} ˙∂_{k} ˙∂_{l}B^{ir} and B^{ir}, defined by (1.2) is written as B^{ir} = (L^{2}/2)g^{ij} − y^{i}y^{j}, we obtain Theorem 1. A Berwald space is conformally closed, if and only if L^{2}g^{ij} are homogeneous polynomials in (y^{i}) of degree two. Remark. Compare the expression of Theorem 1 with that of the definition of Douglas space ([3], p. 388). Both are expressions peculiar to Finsler geometry. Cf. Theorem 7. Example 2. We deal with a Finsler space F^{n} with the m-th root metric L:

L^{m} = a_{hi···l}(x)y^{h}y^{i} ⋯ y^{l},

where the coefficients a_{hi···l}(x) are components of a covariant symmetric m-tensor ([13], [14], [17], [18], [22]). We define covariant (m − r)-tensors

a_{hi···j}(x, y) = a_{hi···jk···l}(x)y^{k} ⋯ y^{l}/L^{r},

and the inverse (a^{ij}) of the matrix (a_{ij}). Then a^{i} = a^{ir}a_{r} is equal to l^{i} = y^{i}/L and

g^{ij} = {a^{ij} + (m − 2)a^{i}a^{j}}/(m − 1).

Consequently Theorem 1 leads to Proposition 3. A Berwald space with a m-th root metric is conformally closed, if and only if L^{2}a^{ij} are homogeneous polynomials in (y^{i}) of degree two. Example 1. (1) We pay attention to the space (R^{3}, L_{1}) again. Putting (y^{i}) = (ẋ, ẏ, ż), we get

(L_{1})^{2}a^{11} = (y^{1})^{2} − 4y^{2}y^{3}, (L_{1})^{2}a^{12} = −y^{1}y^{2} − 2(y^{3})^{2}.

Hence Proposition 3 shows that (R^{3}, L_{1}) is conformally closed as a Berwald space.

(2) We consider the space (R^{n}, L), a generalization of (R^{3}, L_{2}) of Example 1, where L, given by

L^{n} = n!y^{1}y^{2} ⋯ y^{n},

is called the Berwald-Moór metric ([9], Proposition 24.2; [22]). H. Shimada shows

L^{2}a^{11} = −n(n − 2)(y^{1})^{2}, L^{2}a^{12} = ny^{1}y^{2}.

Hence (R^{n}, L) is conformally closed as a Berwald space. Example 3. The ecological metric L ([1], (5.4.1.2); [22]) is given by

L^{m} = (y^{1})^{m} + ⋯ + (y^{n})^{m}.

H. Shimada shows L^{2}a^{11} = L^{m}/(y^{1})^{m−2} and a^{12} = 0. Hence the Minkowski space (R^{n}, L) is not conformally closed as a Berwald space. Example 4. We treat a simple quartic metric L:

L^{4} = 6(y^{1})^{2}y^{2}y^{3}.

It is easy to show typical L^{2}a_{11} = 2y^{2}y^{3}, L^{2}a_{12} = 2y^{1}y^{3}, a_{22} = 0, L^{2}a_{23} = (y^{1})^{2}, and we have

L^{2}a^{11} = −(y^{1})^{2}, L^{2}a^{12} = 2y^{1}y^{2}, L^{2}a^{22} = −4(y^{2})^{2}, L^{2}a^{23} = 2y^{2}y^{3}.

Consequently the Minkowski space (R^{3}, L) is conformally closed as a Berwald space. Example 5. Let us consider a Minkowski space (R^{3}, L) with another quartic metric L:

L^{4} = 6y^{1}y^{2}y^{3}Y, Y = y^{1} + y^{2} + y^{3}.

We have typical L^{2}a_{11} = 2y^{2}y^{3}, L^{2}a_{12} = y^{3}(2Y − y^{3}), and hence

Da^{11} = L^{2}(y^{1})^{2}{4y^{2}y^{3} − (2Y − y^{1})^{2}},
Da^{12} = L^{2}y^{1}y^{2}{(2Y − y^{1})(2Y − y^{2}) − 2y^{3}(2Y − y^{3})},

where D = L^{4}(Y^{2}−y^{1}y^{2}−y^{2}y^{3}−y^{3}y^{1}). Accordingly Proposition 3 shows that (R^{3}, L) is not conformally closed as a Berwald space. Example 6. Finally we consider a Minkowski space (R^{3}, L) with the quartic metric L:

L^{4} = 6(y^{1})^{2}{(y^{2})^{2} + (y^{3})^{2}}.

We have typical L^{2}a_{11} = (y^{2})^{2} + (y^{3})^{2}, L^{2}a_{12} = 2y^{1}y^{2}, L^{2}a_{22} = (y^{1})^{2}, a_{23} = 0, and hence

L^{2}a^{11} = −2(y^{1})^{2}, L^{2}a^{12} = 4y^{1}y^{2},
L^{2}a^{22} = 2{3(y^{3})^{2} − (y^{2})^{2}}, L^{2}a^{23} = −8y^{2}y^{3}.

Therefore (R^{3}, L) is conformally closed as a Berwald space.

3 Conformally closed Berwald spaces, II

We shall write the condition ˙∂_{h} ˙∂_{i} ˙∂_{j}(F g^{ab}) = 0, F = L^{2}/2, stated in Theorem 1 in terms of well-known tensors of F^{n}.

Put F_{hi···k} = ˙∂_{h} ˙∂_{i} ⋯ ˙∂_{k}F. We have F_{hi} = g_{hi} and hence g^{ad}F_{dc} = δ_{c}^{a}, from which we obtain

( ˙∂_{h}g^{ad})F_{dc} + g^{ad}F_{dch} = 0,
( ˙∂_{h} ˙∂_{i}g^{ad})F_{dc} + ( ˙∂_{h}g^{ad})F_{dci} + ( ˙∂_{i}g^{ad})F_{dch} + g^{ad}F_{dchi} = 0,

( ˙∂_{h} ˙∂_{i} ˙∂_{j}g^{ad})F_{dc} + {( ˙∂_{h} ˙∂_{i}g^{ad})F_{dcj} + _{3} } + {( ˙∂_{h}g^{ad})F_{dcij} + _{3} } + g^{ad}F_{dchij} = 0.

Since F_{dcj} = ˙∂_{j}g_{dc} = 2C_{dcj}, the above three equations give respectively

(3.1)  ˙∂_{h}g^{ab} = −2C_{h}^{ab},

(3.2)  ˙∂_{h} ˙∂_{i}g^{ab} = 4(C_{h}^{ar}C_{r i}^{b} + C_{i}^{ar}C_{r h}^{b}) − F _{hi}^{ab},

(3.3)  ˙∂_{h} ˙∂_{i} ˙∂_{j}g^{as} = 2(F _{hi}^{ar}C_{r j}^{b} + F _{hi}^{br}C_{r j}^{a} + _{3} )

−8{(C_{r h}^{a}C_{s i}^{b} + C_{r h}^{b}C_{s i}^{a})C_{j}^{rs} + _{3} } − F _{hij}^{ab},

where F _{hi}^{ab} = g^{ar}g^{bs}F_{rshi} and F _{hij}^{ab} = g^{ar}g^{bs}F_{rshij}.

Let us transvect these equations with y_{a} = g_{ar}y^{r}. (3.1) and (3.2) give respectively y_{a} ˙∂_{h}g^{ab} = 0 and y_{a} ˙∂_{h} ˙∂_{i}g^{ab} = F _{hi}^{b} = 2C_{h i}^{b}. Further (3.3) yields

y_{a} ˙∂_{h} ˙∂_{i} ˙∂_{j}g^{ab} = 2F _{hij}^{b} − 4(C_{h i}^{r}C_{r j}^{b} + _{3} ).

Now we have B_{hij}^{ab} = ˙∂_{h} ˙∂_{i} ˙∂_{j}(F g^{ab}) of the form

(3.4)  B_{hij}^{ab} = F_{hij}g^{ab} + (F_{hi} ˙∂_{j}g^{ab} + F_{h} ˙∂_{i} ˙∂_{j}g^{ab} + _{3} ) + F ˙∂_{h} ˙∂_{i} ˙∂_{j}g^{ab}.

Hence we have

y_{a}B_{hij}^{ab} = 2C_{hij}y^{b} + 2L(l_{h}C_{i j}^{b} + _{3} ) + F {2F _{hij}^{b} − 4(C_{h i}^{r}C_{r j}^{b} + _{3} )}.

In the author’s paper [14] the following equation has been given:

F F_{hijk} = LT_{hijk} − (y_{h}C_{ijk} + _{4} ) + L^{2}C_{hijk}^{2},

C_{hijk}^{2} = C_{hir}C_{j k}^{r} + _{3} ,

where the tensor C^{2}, defined first in [14], is symmetric. Therefore we obtain y_{a}B_{hij}^{ab} = 2LT_{hijr}g^{rb}, and consequently B_{hij}^{ab} = 0 implies T -tensor = 0.

This fact, T -tensor = 0, is certainly in expectation. In fact, first B_{hij}^{ab} = 0 is the necessary and sufficient condition for F^{n} to belong to B_{c}(n), and B_{c}(n) ⊂ L_{c}(n) is clear. Consequently Theorem H shows T -tensor = 0 of F^{n}. We have another reason: (2.1) gives

ȳ_{i}Ḡ_{j kl}^{i} = e^{2c}(y_{i}G_{j kl}^{i} − y_{i}B_{jkl}^{ir}c_{r}).

As it is well-known ([2], §2), F^{n} is a Landsberg space, if and only if y_{i}G_{j kl}^{i} = 0. Therefore the above implies that F^{n} is a conformally closed as a Landsberg space, if and only if y_{i}B_{jkl}^{ir} = 0, and hence Theorem H shows that y_{i}B_{jkl}^{ir} = 0 is equivalent to T -tensor = 0.

Now we shall continue to calculate B_{hij}^{ab} = 0 from(3.4) on the assumption ”T - tensor = 0”. By substituting from(3.1), (3.2) and (3.3), after the long computation, we conclude

(3.5) \frac{1}{2}B_{hij}^{ab} = C_{hij}h^{ab} + (L^{2}C_{h i}^{r}C_{rj}^{2ab} − h_{ij}C_{h}^{ab} + _{3} ) − L^{2}C^{abr}C_{rhij}^{2} − T _{hij}^{ab},

where T _{hij}^{ab} = g^{ar}g^{bs}T_{rshij} is defined in [14] as

(3.6)  2T_{hijkl} = LT_{hijk}|_{l} − (h_{lh}C_{ijk} + h_{li}C_{hjk} + h_{lj}C_{hik} + h_{lk}C_{hij})

+L^{2}(C_{l h}^{r}C_{rijk}^{2} + C_{l i}^{r}C_{rhjk}^{2} + C_{l j}^{r}C_{rhik}^{2} + C_{l k}^{r}C_{rhij}^{2}).

It should be remarked that T_{hijk}, C_{hijk}^{2} and further T_{hijkl} are completely symmetric tensors.

Therefore we can conclude Theorem 2. A Berwald space is conformally closed, if and only if the T -tensor vanishes and

(3.7)  C_{hij}h^{ab} + (L^{2}C_{h i}^{r}C_{rj}^{2ab} − h_{hi}C_{j}^{ab} + _{3} ) − L^{2}C^{abr}C_{rhij}^{2} − T _{hij}^{ab} = 0.

4 Conformally closed Berwald spaces of dimension two

The present section is devoted to conformally closed Berwald spaces of dimension two. Let us apply Berwald’s theory of two-dimensional Finsler spaces in terms of the Berwald frame field (l, m) ([1], [2]). Then the angular metric tensor h_{ij} is written as h_{ij} = ϵm_{i}m_{j} with the signature ϵ = ±1 and the C-tensor is LC_{hij} = Im_{hij} with the main scalar I(x, y), where m_{hij} is the abbreviation of m_{h}m_{i}m_{j}. Cartan’s h-covariant derivatives C_{hij|k} is

LC_{hij|k} = m_{hij}(I_{,1}l_{k} + I_{,2}m_{k}),

where I_{,1}l_{k} + I_{,2}m_{k} = I_{|k} = ∂_{k}I − ( ˙∂_{r}I)G_{k}^{r}.

F^{2} is a Berwald space if and only if I_{,1} = I_{,2} = 0. Then one of the Ricci formulae I_{,1,2}−I_{,2,1} = −RI_{;2}, where L ˙∂_{i}I = I_{;2}m_{i}, yields I_{;2} = 0 or the Gauss curvature R = 0. It is obvious that I_{,1} = I_{,2} = I_{;2} = 0 show I = const., and a Berwald space with R = 0 is locally Minkowski. Consequently we have Berwald’s theorem ([1], Theorem 3.5.3.1; [9], Theorem 28.2): We define five sets of Finsler spaces of dimension two such that

- B(2) ⋯ Berwald spaces,

- B_{1}(2) ⋯ spaces with I = const. and R ≠ 0,

- B_{2}(2) ⋯ spaces with I = const. and R = 0,

- B_{3}(2) ⋯ spaces with I_{;2} ≠ 0 and R = 0,

- M(2) ⋯ locally Minkowski spaces.

Then we have

(4.1)  B(2) = B_{1}(2) ∩ B_{2}(2) ∩ B_{3}(2)  (direct sum),
(4.2)  M(2) = B_{2}(2) ∩ B_{3}(2).

The T -tensor is written as LT_{hijk} = I_{;2}m_{hijk} ([1], [9]). Hence, if we consider the set B_{c}(2) of conformally closed Berwald spaces of dimension two, any F^{2} ∈ B_{c}(2) has I_{;2} = 0 and belongs to B_{1}(2) ∩ B_{2}(2).

We have to pay attention to the second condition (3.7). As has been shown in [14], if I_{;2} = 0 holds, then we have

L^{2}C_{hijk}^{2} = 3ϵI^{2}m_{hijk}, LT_{hijkl} = (6I^{3} − 2ϵI)m_{hijkl}.

As a consequence it is observed that (3.7) holds automatically and we obtain Theorem 3. A Berwald space of dimension two is conformally closed, if and only if it has the constant main scalar, that is,

(4.3)  B_{c}(2) = B_{1}(2) ∩ B_{2}(2).

Remark 1. The main scalar I is conformally invariant. This will suggest Theorem 3. Remark 2. The condition (3.7) does not give any restriction on the assumption ”T - tensor = 0” for the two-dimensional case, as it has been shown above. This remarkable fact is also verified from the ”Reduction theorem of certain Landsberg spaces to Berwald spaces” ([2], [12]): If F^{2} is a Landsberg space with vanishing T -tensor, then F^{2} is a Berwald space. Thus we get B_{c}(2) = L_{c}(2).

5 Conformally closed spaces with cubic metric

We are concerned with the conformal closedness of Finsler spaces with cubic metric L: L^{3} = a_{hij}(x)y^{h}y^{i}y^{j}, where a_{hij} are components of a covariant symmetric 3-tensor. As in Example 2, we put

a_{hi} = a_{hij}(x)y^{j}/L, a_{h} = a_{hij}(x)y^{i}y^{j}/L^{2}.

Then we get [17]

l_{i} = a_{i}, h_{ij} = 2(a_{ij} − a_{i}a_{j}), LC_{hij} = a_{hij} − (a_{hi}a_{i} + _{3} ) + 2a_{h}a_{i}a_{j}.

Throughout the theory of m-th root metrics ([17], [22]) the regularity of the metric, det(a_{ij}) ≠ 0, is assumed. By the inverse (a^{ij}) of the matrix (a_{ij}) we define a^{i} = a^{ir}a_{r} and a_{ij}^{h} = a^{hr}a_{rij}. Then a^{i} = l^{i} and

2LC_{i j}^{h} = a_{ij}^{h} − δ_{i}^{h}a_{i} − δ_{j}^{h}a_{i} + a^{h}(2a_{i}a_{j} − a_{ij}).

The C^{2}-tensor, defined in (3.5) is written in the form

(5.1)  \begin{aligned}2L^{2}C_{hijk}^{2} & = & (a_{hir}a_{jk}^{r} − a_{hi}a_{jk} + _{3} ) \\ & − & 3(a_{h}a_{ijk} + _{4} ) + 4(a_{h}a_{i}a_{jk} + _{6} ) − 12a_{h}a_{i}a_{j}a_{k}. \end{aligned}

As has been shown in [17], the characteristic property of the cubic metric is ˙∂_{h} ˙∂_{i} ˙∂_{j} ˙∂_{k}L^{3} = 0, from which we obtain the theorem: A Finsler space is equipped with cubic metric, if and only if its T -tensor has the form

2LT_{hijk} = −2L^{2}C_{hijk}^{2} − (h_{hi}h_{jk} + _{3} ).

Therefore Theorem H leads to the theorem: A Landsberg space with cubic metric is conformally closed, if and only if

(5.2)  2L^{2}C_{hijk}^{2} + (h_{hi}h_{jk} + _{3} ) = 0.

In the case of the cubic metric we have

h_{hi}h_{jk} + _{3} = 4(a_{hi}a_{jk} + _{3} ) − 4(a_{h}a_{i}a_{jk} + _{6} ) + 12a_{h}a_{i}a_{j}a_{k}.

This together with (5.1) leads (5.2) to the concrete form

(5.2′)  (a_{hir}a_{jk}^{r} + 3a_{hi}a_{jk} + _{3} ) − 3(a_{h}a_{ijk} + _{4} ) = 0.

However the Reduction theorem has been shown in [13]: If F^{n} with cubic metric is a Landsberg space, then it is a Berwald space. Therefore we have finally Theorem 4. A Berwald space with cubic metric is conformally closed, if and only if its T -tensor = 0, that is, (5.2) or (5.2′) holds identically. Remark. If the T -tensor = 0, then from (5.2) we get the following form of T_{hijkl}, defined by (3.6):

2T_{hijkl} = −(h_{hi}C_{jkl} + _{10} ).

Then it is easy to show that the condition (3.7) holds automatically. Example 1. We consider the Minkowski spaces, treated in Example 1 again.

(1) (R^{3}, L_{1}), (L_{1})^{3} = (y^{1})^{3} + (y^{2})^{3} + (y^{3})^{3} − 3y^{1}y^{2}y^{3}.

We have

(L_{1})^{2}a_{11}^{1} = (y^{1})^{2} − 4y^{2}y^{3}, (L_{1})^{2}a_{12}^{1} = {y^{1}y^{3} + 2(y^{2})^{2}}/2,
(L_{1})^{2}a_{22}^{1} = −{y^{1}y^{2} + 2(y^{3})^{2}}, (L_{1})^{2}a_{23}^{1} = −{(y^{1})^{2} − 4y^{2}y^{3}}/2.

Consequently (5.2′) gives T_{1111} = T_{1112} = T_{1122} = T_{1123} = 0, that is, T_{hijk} = 0, h, i, j, k = 1, 2, 3.

(2) (R^{3}, L_{2}), (L_{2})^{3} = y^{1}y^{2}y^{3}.

We have

a_{11}^{1} = a_{22}^{1} = 0, (L_{2})^{2}a_{12}^{1} = y^{1}y^{3}/2, (L_{2})^{2}a_{23}^{1} = −(y^{1})^{2}/2.

Similarly to (1), we get T_{hijk} = 0.

The space (R^{3}, L_{2}) has C_{i} = C_{i r}^{r} = 0 ([9], Proposition 24.2). Similarly (R^{3}, L_{1}) has C_{i} = 0.

The necessary and sufficient condition for a Finsler space with cubic metric to be a Berwald space has been discussed in §2 of [13], but it is a difficult problem to write the condition in terms of the coefficients a_{ijk}(x). Here we write the equation

La_{hr}G_{i jk}^{r} + (a_{hir}G_{j k}^{r} + _{3} ) = {ijk, h}

where G_{i jk}^{r} is the hv-curvature tensor of the Berwald connection and {ijk, h} are generalized Christoffel symbols, defined first in [18].

6 Conformal closedness of Berwald spaces with (α, β)-metric

We consider a generalized Randers space F^{n} [20], which is a Finsler space with (α, β)- metric L(α, β) ([1], [9]). If we deal with a conformal change of metric (1.1), then we have

L̄ = e^{c}L(α, β) = L(e^{c}α, e^{c}β),

because L(α, β) is assumed to be positively homogeneous in (α, β) of degree one. Consequently we get L̄ = L(ᾱ, β̄), (ᾱ, β̄) = (e^{c}α, e^{c}β) [6], and hence the changed space F̄^{n} is still a generalized Randers space. Therefore, similarly to the case of the 1-form metric, we have Proposition 4. The notion of the generalized Randers space is conformally closed.

We now consider a Randers space F^{n} with L = α + β. Theorem 4 of the paper [8] states that if F^{n} has the T -tensor = 0, then β should vanish, that is, F^{n} is reduced to a Riemannian space with the Riemannian metric α. Therefore Theorem 2 leads to Theorem 5. If a Randers space is conformally closed Berwald space, then it is a Riemannian space.

Here we shall show a direct proof of this theorem. A Randers space F^{n} with L = α + β, α^{2} = a_{ij}(x)y^{i}y^{j}, β = b_{i}(x)y^{i}, is a Berwald space ([7], [8]), if and only if b_{i}_{;j} = 0 in the Levi-Civita connection {γ_{j k}^{i}(x)} of the associated Riemannian space with α. The conformally changed space F̄^{n} of F^{n} has ā_{ij} = e^{c}a_{ij} and b̄_{i} = e^{c}b_{i}. From γ̄_{j k}^{i} = γ_{j k}^{i} + δ_{j}^{i}c_{k} + δ_{k}^{i}c_{j} − c^{i}a_{jk}, c_{k} = ∂_{k}c, c^{i} = a^{ir}c_{r} we have

(6.1)  b̄_{i}_{;j} = e^{c}(b_{i}_{;j} − c_{i}b_{j} + b_{r}c^{r}a_{ij}).

If both F^{n} and F̄^{n} are Berwald spaces, then (6.1) gives c_{r}(b^{r}a_{ij} − δ_{i}^{r}b_{j}) = 0. If this is satisfied for any c(x), then we have b^{r}a_{ij} = δ_{i}^{r}b_{j}, which implies b_{i} = 0. Thus we proved Theorem 5.

Secondly we consider a Kropina space F^{n} with L = α^{2}/β. Theorem 2 of C. Shibata’s paper [21] states that the T -tensor of F^{n} never vanishes. Therefore we have Proposition 5. A Kropina space is not a conformally closed Berwald space.

We shall treat a conformal change of a Kropina space of Berwald type in detail. A Kropina space F^{n} is a Berwald space ([7], [10], [21]) if and only if we have function f_{i}(x) satisfying

(6.2)  b_{i}_{;j} = f_{r}b^{r}a_{ij} + b_{i}f_{j} − b_{j}f_{i}, b^{r} = a^{ri}b_{i}.

The conformally changed Kropina space F̄^{n} is also assumed to be a Berwald space:

b̄_{i}_{;j} = f̄_{r}b̄^{r}ā_{ij} + b̄_{i}f̄_{j} − b̄_{j}f̄_{i} = e^{c}(f̄_{r}b^{r}a_{ij} + b_{i}f̄_{j} − b_{j}f̄_{i}).

We have (6.1) and (6.2), and hence the above gives

(f̄_{r} − f_{r} − c_{r})b^{r}a_{ij} = (f̄_{i} − f_{i} − c_{i})b_{j} − (f̄_{j} − f_{j})b_{i}.

This yields (2f̄_{i} − 2f_{i} − c_{i})b_{j} = (2f̄_{j} − 2f_{j} − c_{j})b_{i}, so that we have a function κ(x) satisfying 2f̄_{i} − 2f_{i} − c_{i} = 2κb_{i}. Hence (c_{r} − 2κb_{r})b^{r}a_{ij} = b_{i}c_{j} + b_{j}c_{i}, and n = 2 is necessary, provided (c_{r} − 2κb_{r})b^{r} ≠ 0. In the case of n > 2 we have c_{i} = 0. Therefore we have Theorem 6. Let a Kropina space F^{n}, n > 2, be a Berwald space. The conformally changed Kropina space F^{n} is still a Berwald space, if and only if the change is homothetic.

7 Conformal change of Douglas spaces

A Finsler space is by definition a Douglas space ([3], [4]), if D^{ij} = G^{i}y^{j} − G^{j}y^{i} are homogeneous polynomials in (y^{i}) of degree three. Definition. A Douglas space is called conformally closed, if it remains to be a Douglas space by any conformal change of metric.

For a conformal change (1.1) we have (1.3). Thus, for D^{ij} we get D̄^{ij} = D^{ij} − D^{ijr}c_{r}, where D^{ijr} = B^{ir}y^{j} − B^{jr}y^{i}. From (1.2) we have

(7.1)  D^{ijr} = F (g^{ir}y^{j} − g^{jr}y^{i}), F = L^{2}/2.

Therefore we have Theorem 7. A Douglas space is conformally closed, if and only if D^{ijr} of (7.1) are homogeneous polynomials in (y^{i}) of degree three.

It is shown [3] that a Finsler space is a Douglas space, if and only if the projective invariants Q^{i} = G^{i} − G_{r}^{r}y^{i}/(n + 1) are homogeneous polynomials in (y^{i}) of degree two. (1.3) and (1.4) show

Q̄^{i} = Q^{i} − {B^{ir} − B_{s}^{rs}y^{i}/(n + 1)}c_{r}.

Consequently we have Proposition 6. A Douglas space is conformally closed, if and only if B^{ij} −B_{r}^{jr}y^{i}/(n+ 1) are homogeneous polynomials in (y^{i}) of degree two.

We treat a Kropina space with L = α^{2}/β again. Since a two-dimensional Kropina space is Douglas space without any restriction [3], we have immediately Theorem 8. A two-dimensional Kropina space is a conformally closed Douglas space.

For a Kropina space of arbitrary dimension we have ([9], [21])

2(α/β)^{2}g^{ij} = a^{ij} − b^{i}b^{j}/b^{2} + (2β/b^{2}α^{2})(b^{i}y^{j} + b^{j}y^{i}) + (2/b^{2}α^{4})(b^{2}α^{2} − 2β^{2})y^{i}y^{j}.

Hence we have

2D^{ijk} = {α^{2}a^{ik}/2 − (α^{2}/2b^{2})b^{i}b^{k} + (β/b^{2})b^{i}y^{k}}y^{j} − [i, j],

where [i, j] denotes the interchange of i, j. Thus Theorem 7 yields Theorem 9. If a Kropina space F^{n}, n > 2, is Douglas space, then it is a conformally closed Douglas space.

If we consider a conformal change of an (α, β)-metric, then we get L̄ = e^{c}L(α, β) = L(ᾱ, β̄), where

ᾱ = e^{c}α (ā_{ij} = e^{c}a_{ij}); β̄ = e^{c}β (b̄_{i} = e^{c}b_{i}).

For s_{ij} = (∂_{j}b_{i} − ∂_{i}b_{j})/2 we get

(7.2)  s̄_{ij} = e^{c}{s_{ij} + (b_{i}c_{j} − b_{j}c_{i})/2}.

We are concerned with the conformal change of a Randers space F^{n} with L = α+β. F^{n} is a Douglas space [3], if and only if s_{ij} = 0, that is, there exists locally a function b(x) such that b_{i} = ∂_{i}b. Then (7.2) is reduced to s̄_{ij} = e^{c}(b_{i}c_{j} − b_{j}c_{i})/2. Hence we get s̄_{ij} = 0, that is, F̄^{n} is still a Douglas space, if and only if b_{i}c_{j} − b_{j}c_{i} = 0, that is, c_{i} is proportional to b_{i}. Therefore we have Theorem 10. Let F^{n} be a Randers space of Douglas type with L = α + β, provided that β = (∂_{i}b(x))y^{i} ≠ 0.

(1) The conformally changed F̄^{n} is not of Douglas type in general.

(2) F̄^{n} is also of Douglas type, if and only if the conformal factor c(x) is such that ∂_{i}c is proportional to ∂b_{i}.

From (7.2) it is follows that if we put s_{i} = b^{r}s_{ri}, then we get

s̄_{i} = s_{i} + (b^{2}c_{i} − b^{r}c_{r}b_{i})/2, b^{2} = b^{r}b_{r},
s̄_{ij} − (b̄_{i}s̄_{j} − b̄_{j}s̄_{i})/b̄^{2} = e^{c}{s_{ij} − (b_{i}s_{j} − b_{j}s_{i})/b^{2}}.

Consequently, if F^{n} has s_{ij} − (b_{i}s_{j} − b_{j}s_{i})/b^{2} = 0, so is the conformally changed F̄^{n}. The condition s_{ij} −(b_{i}s_{j} −b_{j}s_{i})/b^{2} = 0 is necessary and sufficient for a Kropina space F^{n}, n > 2, to be a Douglas space [16]. Therefore we could obtain another proof of Theorem 9 not due to Theorem 7.

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