Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 143–149

GLOBAL GEODESIC MAPPINGS AND THEIR GENERALIZATIONS

FOR COMPACT RIEMANNIAN SPACES

Josef Mikeš

Abstract. This paper is devoted to the study of global geodesic mappings and their generalizations for compact Riemannian spaces both with boundary and without boundary. Key words: Geodesic mappings, compact Riemannian spaces. MS classification: 53C20.

1. Introduction

General questions of global geodesic mappings have been studied by G. Vranceanu [16] and many other authors [2], [6], [7], [9], [14], …

Conditions on compact Riemannian spaces V_{n} without boundary which do not admit global nontrivial geodesic mappings can be found in [2], [6], [14], [16]. We have succeeded in showing the existence of the compact Riemannian spaces V_{n} without boundary, admitting global nontrivial geodesic mapping [7], [9].

This paper is devoted to the study of global geodesic mappings and their generalizations for compact Riemannian spaces both with boundary and without boundary.

Throughout the paper, all the objects are assumed to be sufficiently smooth, no conditions are laid down on the signature of the metric ds^{2} of a Riemannian space V_{n}, otherwise they are explicitly indicated.

A diffeomorphism of a Riemannian space V_{n} onto V̄_{n} which maps all geodesics in V_{n} into geodesics in V̄_{n} is called a geodesic mapping [3]. A geodesic mapping which is not affine is said to be nontrivial.

A diffeomorphism f: V_{n} → V̄_{n} is a geodesic mapping if and only if, for every vector field x on V_{n}, we have

¯∇_{x}x − ∇_{x}x = 2ψ(x)x  (1)

where ∇, or ¯∇ denotes the Riemannian connections of V_{n}, or the connection induced on V_{n} by the Riemannian connection of V̄_{n}, ψ(x) is a linear (differential) form on V_{n}.

In the next pages, we shall often identity the geometric objects on V̄_{n} with the corresponding objects on V_{n} induced by a fixed diffeomorphism f: V_{n} → V̄_{n}. If ψ(x) = 0, then the mapping is known to be affine, otherwise this geodesic mapping is nontrivial.

In [10] the concept of F -planar mapping was introduced. Consider a special F -planar mapping of a Riemannian space V_{n} into a Riemannian space V̄_{n}, characterized by the

Typeset by AMS-TEX

This paper is in final form and no version of it will be submitted for publication elsewhere.

following conditions:

\begin{matrix}a) & ¯∇_{x}x − ∇_{x}x = 2ψ(x)x + 2φ(x)F (x); \\ b) & g(x, F (x)) = 0; \\ c) & ḡ(x, F (x)) = 0 \end{matrix}  (2)

where ψ, φ are linear forms, F is a (1,1) tensor field determined on V_{n}, g and ḡ are the metric forms of V_{n} and V̄_{n}. Such a mapping will be called F^{∗}-planar.

If φ = 0 then a F^{∗}-planar mapping is a geodesic mapping. Special cases of F^{∗}-planar mappings are known as holomorphically-projective mappings of Kählerian spaces [16]. We note that the conditions (2a,c) hold for quasigeodesic mappings, introduced by A.Z. Petrov [11].

2. Global geodesic and F^{∗}-planar mappings V_{n} into V̄_{n} with boundary ∂V

In this part we suppose that the metric of the space V_{n} is positive definite.

Theorem 1. Let an orientable compact Riemannian space V_{n} (n ≥ 2) admit a global geodesic mapping onto V̄_{n}. If all sectional curvatures at each point m ∈ V_{n} are nonpositive, and the metrics of V_{n} and V̄_{n} obey the relation ḡ(x, y) = f(m)g(x, y) at each point m ∈ ∂V, then the geodesic mapping V_{n} onto V̄_{n} is trivial, i.e. it is affine.

This theorem is generalized by means of

Theorem 2. Let an orientable compact Riemannian space V_{n} (n ≥ 4) admit a global F^{∗}-planar mapping onto V̄_{n} with ψ(x) being a gradient. If all sectional curvatures at each point m ∈ V_{n} are nonpositive, and the metrics of V_{n} and V̄_{n} obey the condition ḡ(x, y) = f(m)g(x, y) at each point m ∈ ∂V, then the F^{∗}-planar mapping of V_{n} onto V̄_{n} is affine.

In particular, Theorem 2 is true for holomorphically-projective mappings of Kählerian spaces. In case of ∂V = ∅ the condition on ∂V is satisfied, and the signature of the metric ḡ(x, y) may be arbitrary.

The proofs of the above mentioned theorems are based on the following assertion.

Theorem 3. Let V_{n} be an orientable compact Riemannian space and a(x, y) be a symmetric bilinear form on V_{n}, let a function J be determined in the following way

J ≡ g^{ij}g^{kl}g^{mp}(a_{ik,l}a_{jm,p} − a_{ik,m}a_{jp,l})  (3)

where a_{ij} are local coordinates of a(x, y), ∥g^{ij}∥ = ∥g_{ij}∥^{−1}, g_{ij} are local components of the form g(x, y), and comma denotes the covariant derivative with respect to the connection on V_{n}. Then, if

1) all sectional curvatures at each point m ∈ V_{n} are nonpositive,

2) the function J ≥ 0 at each point of V_{n},

3) there exists a function f: ∂V → R such that a(x, y) = f(m)g(x, y) at each point m ∈ ∂V,

then J = 0 at each point m ∈ V_{n}.

In case that of all the sectional curvatures of V_{n} are negative, then under the conditions 2) and 3) there exists a function f such that a(x, y) = f(m)g(x, y) at each point m ∈ V_{n}.

Theorem 3 is analogous to that of A. Švec, and it can be verified following [1,p. 12]. Let us restrict our attention to the proof of Theorem 2. Suppose that the form ψ is

the differential of a function Ψ, that is ψ(x) = ∇_{x}Ψ. Let us put

a^{ij} ≡ exp(2Ψ) ḡ^{ij}  (4)

where ∥ḡ^{ij}∥ = ∥ḡ_{ij}∥^{−1}, and ḡ_{ij} are local components of the form ḡ(x, y).

It is easy to show that (2a) leads to the following relations

a_{,k}^{ij} = λ^{(i}δ_{k}^{j)} + Θ^{(i}F_{k}^{j)}

where

λ^{i} = − exp(2Ψ) ḡ^{iα}ψ_{α}; Θ^{i} = − exp(2Ψ) ḡ^{iα}φ_{α},

ψ_{i}, φ_{i}, F_{k}^{j} are local coordinates of ψ(x), φ(x) and F (x) respectively, δ_{i}^{h} is the Kronecker symbol, (i, j) denotes symmetrization with respect to i and j.

Putting

a_{ij} ≡ a^{αβ}g_{iα}g_{jβ}  (5)

we can verify that the invariant J may be written in the following way:

J = −∥Θ^{h}∥^{2}F_{β}^{α}F_{α}^{β} + \frac{1}{2}∥2Θ^{α}F_{α}^{h} + (n + 2) λ^{h}∥^{2} + \frac{1}{2}(n^{2} − 2n − 8) ∥λ^{h}∥^{2}

where ∥ξ∥^{2} ≡ ξ^{α}ξ^{β}g_{αβ} is the square of the norm of the vector ξ^{h}.

It is obvious that for n ≥ 4 it holds J ≥ 0.

The conditions at the boundary of ∂V: ḡ(x, y) = f(m)g(x, y) in virtue of (4) and (5) generate the condition

a(x, y) = f(m) exp(2Ψ) g(x, y).

Hence, all the conditions of Theorem 3 are satisfied and J = 0. By a direct analysis we can check that J = 0 if and only if λ^{h} = Θ^{h} = 0. The latter is equivalent to ψ(x) = φ(x) = 0. Hence, the F^{∗}-planar mapping is affine, and Theorem 2 has been proved.

Theorem 4. Let an orientable compact Riemannian space V_{n} admit a global affine mapping onto V̄_{n}. If all the sectional curvatures at each point m ∈ V_{n} are positive (or negative), and the metrics of V_{n} and V̄_{n} are connected by the relations ḡ(x, y) = = f(m)g(x, y) at each point m ∈ ∂V, then the mapping of V_{n} into V̄_{n} is a homothety.

Proof. A Riemannian space V_{n} admits an affine mapping onto V̄_{n} if and only if ∇_{z}ḡ(x, y) = 0. Putting a(x, y) = ḡ(x, y) we assert that J = 0. Then on the basis of A. Švec’s theorem [1, p. 12] and in virtue of Theorem 3 it follows ḡ(x, y) = f(m)g(x, y) at each point m ∈ V_{n}. Obviously, the function f = const, and the proof is finished.

3. Global geodesic mappings of V_{n}(B)-spaces

A Riemannian space V_{n} is called a V_{n}(B)–space if it satisfies the following conditions

\begin{matrix}a) & ∇_{z}a(x, y) = λ(x)g(y, z) + λ(y)g(x, z); \\ b) & ∇_{y}λ(x) = μ g(x, y) + B a(x, y) \end{matrix}  (6)

where a(x, y) is a symmetric regular quadratic form, λ(x) is a nonvanishing linear form, and μ, B are functions.

If B = const, then the conditions ∇_{x}μ = 2B λ(x) are satisfied. If the relations (6) are true in all points, then V_{n}(B) is globally determined. By the correctness of the equations (6a) under the condition λ(x) ≢ 0 the existence of a nontrivial geodesic mapping is guaranteed [13].

Theorem 5. There exists no compact space V_{n}(B), for B = 0.

The proof of this theorem follows from the fact that the relation (6b) may be written in the form

∇_{x}∇_{y}λ = μ g(x, y)  (7)

where μ = const, λ(x) = ∇_{x}λ.

Vector fields, satisfying locally (7) were studied by P.A. Shirokov [12]. We and S.G. Leiko have shown that the equations (7) on a compact V_{n} have only a trivial solution λ(x) = ∇_{x}λ = 0.

Note. It is shown in [6] that in a compact pseudo Riemannian space V_{n} the equation ∇_{x}∇_{y}λ = μ g(x, y) with μ being a function has the solutions λ = const and μ = 0.

Theorem 6. Let a compact V_{n}(B), B = const, admit a global geodesic mapping onto V̄_{n}. Then V̄_{n} is V̄_{n}(B̄) with B̄ = const, and the signatures of the metrics of V_{n}(B) and V̄_{n}(B̄) coincide.

In case V_{n}(B) is a Riemannian space with a positive signature of the metric, Theorem 6 has been proved by V.A. Kiosak [2].

It has been shown that the spaces of constant curvature, the Einstein spaces, the generalized semisymmetric (pseudosymmetric), generalized Ricci-semisymmetric (Riccipseudosymmetric) spaces, admitting nontrivial geodesic mapping may be, in particular, the spaces V_{n}(B) [4], [5]. Hence, for all the spaces mentioned above Theorems 5 and 6 can be applied.

4. Geodesic mapping of generalized T -semisymmetric spaces

For generalized T -semisymmetric spaces determined in [5] the following assertion may be proved.

Theorem 7. Let a tensor field T of (0, m) – type exist in a Riemannian space V_{n} (n > 2(m − 1)) and let the following condition be satisfied

∑_{r=1}^{m} T_{i_{1}i_{2}...i_{r−1}αi_{r+1}...i_{m}} Z_{i_{r}kl}^{α} = 0  (8)

where Z_{ijk}^{h} ≡ R_{ijk}^{h} + B(δ_{k}^{h}g_{ij} − δ_{j}^{h}g_{ik}), R_{ijk}^{h} is the Riemann tensor. If V_{n} admits a geodesic mapping, then either V_{n} is V_{n}(B) or the tensor T may be represented as the tensor sum of products of invariants and the metric tensor.

Proof. Let in V_{n} the pointed tensor T exist, satisfying the conditions (8). Further, we shall suppose V_{n} to admit a geodesic mapping. Hence by [1],[5] in V_{n} the equations (6a) hold. The conditions of their integrability are

a_{α(i}R_{j)kl}^{α} = λ_{l,(i}g_{j)k} − λ_{k,(i}g_{j)l}.  (9)

where a_{ij} and λ_{i} are local coordinates of the forms a(x, y) and λ(x) respectively. In [5] the theorem is proved for the case when m = 1, 2, and a special case when m = 4 and T is the Riemann tensor. By induction we shall prove the theorem for an arbitrary m > 2 assuming that it is correct for 1,2,...,m–1. Let us suppose the converse, i.e. that for a geodesic mapping the conditions (6) do not hold. Evidently, any convolution of T with g^{ij} satisfies the relations (8) and by our assumption these convolutions are a tensorial sum of products of the invariants and the metric tensor. For the m odd the pointed convolutions are equal to zero. For m even and n ≥ 2m − 1 one easily can construct a tensor T^{∗}, as a sum of the tensor T and the tensorial sums of products of the definite invariants and the metric tensor, such that any convolution of T^{∗} with g^{ij} is zero. The tensor T^{∗} satisfies the conditions (8), therefore without loss of generality, later on we will consider the tensor T^{∗} instead of T.

We shall convolve the relation (8) with a_{kα}g^{αj}, and then symmetrize with respect to l and k. Taking into account the structure of the tensor Z_{ijk}^{h}, the conditions (9) and the properties of the Riemann tensor, after a corresponding grouping we get

∑_{r=1}^{m}{Λ_{h_{r}(i}T_{|h_{1}...h_{r−1}|j)h_{r+1}...h_{m}} − g_{h_{r}(i}T_{|h_{1}...h_{r−1}|j)h_{r+1}...h_{m}}^{r}} = 0  (10)

where

Λ_{ij} ≡ λ_{i,j} − μ g_{ij} − B a_{ij};  (11)
μ ≡ (λ_{α,β} − Ba_{αβ})g^{αβ}/n;
T_{h_{1}...h_{m}}^{r} ≡ Λ_{αh_{r}}g^{αβ}T_{h_{1}...h_{r−1}βh_{r+1}...h_{m}}.

Convolving (10) in turn with g^{ih_{p}} g^{jh_{q}} where p, q = 1, 2, ..., m (p ≠ q), we shall obtain equations which for n > 2(m − 1) lead to

g^{αβ}T_{h_{1}...h_{p−1}αh_{p+1}...h_{r−1}βh_{r+1}...h_{m}}^{r} = 0.  (12)

Then we convolve (10) with g^{ih_{p}}. From the obtained relations, taking into account (12) we shall convince, that T_{h_{1}...h_{m}}^{r} = 0. Then the formulas (10) are simplified. From them one can show that either Λ_{ij} = 0, or T_{h_{1}...h_{m}} = 0. Taking into account the formulas (11), the theorem is completely proved.

It is easy to check that the following assertion is true. By Theorem 7, and taking into account [8] the local result follows:

Theorem 8. A Riemannian space with nonconstant curvature V_{n} and with one of the following relations being satisfied

a) R_{ijk,m_{1}m_{2}...m_{p}}^{h} = 0, (n > 2(p + 1))

or

b) R_{i[j,k],m_{1}m_{2}...m_{p}} = 0, R_{i[j,k]} ≠ 0 (n > 2p)

where R_{ij} is the Ricci tensor, [j, k] denotes the alternation with respect to j and k, does not admit a local nontrivial geodesic mapping.

Theorem 9. Let in a compact Riemannian space V_{n} ∈ C^{3} (n ≥ 2) one of the following relation be true at each point:

1) T_{h_{1}h_{2}...h_{p},[lm]} = 0 and the tensor T cannot be represented as a sum of products of invariants and the metric tensor (n > 2(p − 1));

2) R_{ij,[lm][pr]} = 0. If the condition 1) is satisfied at least at one point then V_{n} does not admit a global

geodesic mapping onto V̄_{n} ∈ C^{3}.

Theorem 10. Suppose that in a compact Riemannian space V_{n} ∈ C^{k} (n ≥ 2) one of the following conditions is satisfied:

a)  R_{ij,[lm][pr]} = 0; R_{ij,[lm]} ≢ 0;  (k = 6);
b)  R_{ij,[lm]} = 0;  R_{ij} ≢ ρ g_{ij};  (k = 4);
c)  R_{ijk,[lm]}^{h} = 0;  R_{ijk}^{h} ≢ ρ(δ_{j}^{h}g_{ik} − δ_{k}^{h}g_{ij}); (k = 4);
d)  R_{ij} = 0;  (k = 3).

Then the Riemannian space V_{n} does not admit a global nontrivial geodesic mapping onto V̄_{n} ∈ C^{3}.

References

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Josef Mikeš, Odessa university, Petra Velikogo 2, Odessa, Ukraine

Received 8 October 1992 Revised 30 January 1993