PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 94 (108) (2013), 125–130 DOI: 10.2298/PIM1308125H

GEODESIC MAPPINGS ON COMPACT RIEMANNIAN MANIFOLDS WITH CONDITIONS ON SECTIONAL CURVATURE

Irena Hinterleitner

Abstract. We found new criteria for sectional curvatures on compact Riemannian manifolds for which geodesic mappings are affine, and, moreover, homothetic.

1. Introduction

To the theory of geodesic mappings and their transformations have been devoted many papers, these results are formulated in a large number of research papers and monographs [2, 4–12, 16–19, 21–26, 30, 33], etc.

In 1953, Takeno and Ikeda [31] considered geodesic mappings of spherically symmetric spaces V_{4}, in 1954 Sinyukov [26, p. 88] studied the case of symmetric and recurrent spaces and, in 1976 Mikeš ( [13,16], [21, p. 206], [26, pp. 151–155]) proved that generalized recurrent (pseudo-) Riemannian spaces V_{n} with nonconstant curvature do not admit nontrivial geodesic mappings. In this topic Prvanović [23] and Sobchuk [20, 29] also have been interested. These results were obtained “locally" and they are contained in [14, 16, 21, 26].

Global results for geodesic mappings of compact Riemannian manifolds were obtained by Vrançeanu [33], Sinyukova [27, 28], Mikeš [15, 16], etc.

The above results are related to questions of projective rigidity of (pseudo-) Riemannian manifolds and also of manifolds with affine connections.

In [10] and [11] we proved that these mappings preserve the smoothness class of metrics of geodesically equivalent (pseudo-) Riemannian manifolds. In [10] it was sufficient to suppose the metrics to be of differentiability class C^{2}, and in [11] to be of class C^{1}.

We present new results on geodesic mappings of compact Riemannian manifolds with certain conditions on the sectional curvature of the Ricci directions.

2010 Mathematics Subject Classification: Primary 53C20; Secondary 53C21, 53C24.

Key words and phrases: geodesic mapping, affine mapping, homothetic mapping, compact Riemannian manifold.

The paper was supported by the project FAST-S-13-2088 of the Brno University of Technology.

2. Geodesic mapping theory

Let V_{n} = (M, g) and V̄_{n} = (M̄, ḡ) be n-dimensional (pseudo-) Riemannian manifolds with metrics g and ḡ, respectively.

Definition 2.1. A diffeomorphism f : V_{n} → V̄_{n} is called a geodesic mapping of V_{n} onto V̄_{n} if f maps any geodesic in V_{n} onto a geodesic in V̄_{n}.

We restricted ourselves to the study of a coordinate neighborhood (U, x) of the points x ∈ V_{n} and f(x) ∈ V̄_{n}. The points x and f(x) have the same coordinates x = (x^{1}, …, x^{n}). We assume that V_{n}, V̄_{n} ∈ C^{1} (g, ḡ ∈ C^{1}) if their components g_{ij} (x), ḡ_{ij} (x) ∈ C^{1} on (U, x), respectively.

It is known [12], see [6, pp. 131–133], [21, p. 167], that V_{n} admits a geodesic mapping onto V̄_{n} if and only if the following Levi-Civita equations

(2.1)  ∇_{k} ḡ_{ij} = 2ψ_{k} ḡ_{ij} + ψ_{i} ḡ_{jk} + ψ_{j} ḡ_{ik}

hold, where ∇ is the Levi-Civita connection on V_{n} and

ψ_{i} = ∂_{i}Ψ, Ψ = \frac{1}{n + 1} ln \sqrt{| det ḡ/ detg|}, ∂_{i} = ∂/∂x^{i}.

Sinyukov [26, p. 121], see [21, p. 167], proved that the Levi-Civita equations (2.1) are equivalent to

(2.2)  ∇_{k}a_{ij} = λ_{i}g_{jk} + λ_{j} g_{ik} ,

where

(2.3)  (a) a_{ij} = e^{2Ψ} ḡ^{αβ} g_{iα} g_{jβ} ; (b) λ_{i} = −e^{2Ψ} ḡ^{αβ} ψ_{α} g_{iβ} ,

and, moreover, λ_{i} = ∂_{i}Λ, Λ = \frac{1}{2} a_{αβ} g^{αβ} . Here (ḡ^{ij} ) = (ḡ_{ij} )^{−1} and (g^{ij} ) = (g_{ij} )^{−1} . On the other hand:

ḡ_{ij} = e^{2Ψ} ĝ_{ij} , Ψ = ln \sqrt{| det ĝ/ detg|}, (ĝ_{ij} ) = (a_{αβ} g^{iα} g^{jβ} )^{−1} .

Furthermore, we assume that V_{n} = (M, g) ∈ C^{2} and V̄_{n} = (M, ḡ) ∈ C^{2}. In this case, the integrability conditions of the equations (2.2), due to the Ricci identity

(2.4)  ∇_{l}∇_{k}a_{ij} − ∇_{k}∇_{l}a_{ij} = a_{iα} R_{jkl}^{α} + a_{jα} R_{ikl}^{α} ,

have the following form

(2.5)  a_{iα} R_{jkl}^{α} + a_{jα} R_{ikl}^{α} = g_{ik} ∇_{l}λ_{j} + g_{jk} ∇_{l}λ_{i} − g_{il} ∇_{k}λ_{j} − g_{jl} ∇_{k}λ_{i},

where R_{ijk}^{h} are components of the Riemannian tensor R on V_{n}, and after contraction with g^{ik} we get [26, p. 133]

(2.6)  n∇_{l}λ_{j} = μ g_{jl} − a_{jα} R_{l}^{α} − a_{αβ} R_{j} ^{αβ} _{l},

where μ = ∇_{α} λ^{α}, R_{i}^{α} = g^{αβ} R_{βi} and R_{ij} = R_{iαj}^{α} are components of the Ricci tensor Ric on V_{n}.

3. Integral formula

We introduce the vector field ξ on V_{n} ∈ C^{2} in the following way

(3.1)  ξ^{i} = a_{β}^{α} ∇_{α} a^{iβ} − a_{β}^{i}∇_{α} a^{αβ} ,

where a_{i}^{l} = g^{lα} a_{αi} , a^{ij} = a_{αβ} g^{iα} g^{jβ} . Using formula (3.1), the Ricci identity (2.4) and Sinyukov’s equations (2.2) we obtained that the divergence of the vector ξ has the following representation

divξ = Φ(a) − (n − 1)(n + 2) λ_{α} λ_{β}g^{αβ} ,

where Φ(a) = R_{ij} a^{ik} a_{k}^{j} − R_{ijkl} a^{ik} a^{jl} .

Suppose that the Riemannian manifold (M, g) is compact and without boundary, then on the basis of the Gauß theorem ∫_{M} divξ dν = 0 we obtain the integral formula

(3.2)  ∫_{M} Φ(a) dν = (n − 1)(n + 2) ∫_{M} λ_{α} λ_{β}g^{αβ} dν.

For applying the Gauss theorem it is necessary to require the orientability of M, if M is a non-orientable manifold, then we’ll look at the oriented double cover.

Let g(e_{i}, e_{j}) = δ_{ij} and a(e_{i}, e_{j}) = α_{i}δ_{ij} with the Kronecker symbol δ_{ij} , i.e., {e_{1}, …, e_{n}} is the orthonormal basis of eigenvectors to the eigenvalues α_{1}, …, α_{n} of the tensor a = (a_{ij} ) of T_{x}M at any point x ∈ M. As we can see from direct calculation, Φ(a) has the following form (see [3, p. 592]):

(3.3)  Φ(a) = ∑_{i<j} K(e_{i}, e_{j})(α_{i} − α_{j})^{2},

where K(e_{i}, e_{j}) are sectional curvatures in the two-directions e_{i} ∧ e_{j}.

It is easy to see:

Φ(a) = R_{ij} a^{ik} a_{k}^{j} − R_{ijkl} a^{ik} a^{jl} = ∑_{i,j} (α_{i})^{2}R_{ijij} − ∑_{i,j} α_{i}α_{j} R_{ijij}

= ∑_{i<j} ((α_{i})^{2} + (α_{j})^{2}) · R_{ijij} − 2 ∑_{i<j} α_{i}α_{j} R_{ijij}

= ∑_{i<j} (α_{i} − α_{j})^{2} · R_{ijij} = ∑_{i<j} (α_{i} − α_{j})^{2} · K(e_{i}, e_{j}),

where

K(e_{i}, e_{j}) = \frac{R(e_{i}, e_{j} , e_{i}, e_{j})}{g(e_{i}, e_{i}) · g(e_{j} , e_{j}) − (g(e_{i}, e_{j}))^{2}} = R_{ijij} .

4. Principal orthonormal basis

Eisenhart [6, pp. 113–114] introduced a principal direction in a Riemannian manifold (M, g), as an eigenvector of the Ricci tensor. He showed that at any point x ∈ M there exists the orthonormal basis {e_{1}, …, e_{n}} in which

g_{ij} = δ_{ij} and R_{ij} = ρ_{i}δ_{ij} ,

i.e., e_{1}, …, e_{n} are the vectors of the principal directions and ρ_{1}, …, ρ_{n} are their eigenvalues. This basis is called the principal orthonormal basis.

This means that the existence of this basis is a property only of the Riemannian manifold (M, g), independent of the solution a_{ij} of equation (2.2). Generally the set of principal orthonormal bases is a proper subset of the set of orthonormal bases. Because the vector field λ_{i} is gradient-like, formula (2.6) implies [26, p. 138]

a_{iα} R_{j}^{α} = a_{jα} R_{i}^{α} .

So the tensors a_{ij} and R_{ij} commute and have common eigenvectors. From this fact it follows that there exist a principal orthonormal basis in which g_{ij} = δ_{ij} and a_{ij} = α_{i}δ_{ij} hold. This basis is called a joint principal orthonormal basis. Note that we do not restrict the signature of the Ricci tensor and the tensor a_{ij} . In the following we restrict ourselves to the study of formulas (3.2) and (3.3) on joint principal orthonormal bases.

5. Main Theorems

For the following we recall that a compact Riemannian manifold V_{n} admits a geodesic mapping onto a (pseudo-) Riemannian manifold V̄_{n}.

If we assume that at each point x ∈ M all sectional curvatures K(e_{i}, e_{j}) are non-positive in the two-directions e_{i} ∧ e_{j} of the joint principal orthonormal basis {e_{1}, …, e_{n}} of vectors of the main directions of the Ricci tensor, then from integral formula (3.2) it follows

(5.1)  (a) ∫_{M} Φ(a) dν = 0 and (b) ∫_{M} λ_{α} λ_{β}g^{αβ} dν = 0.

From integral (5.1b) follows λ_{α} λ_{β}g^{αβ} = 0 and this fact implies that λ_{i} is vanishing on M, i.e., λ_{1} = ⋯= λ_{n} = 0. In this case, the geodesic mapping is affine (see [21, p. 150]). We proved the following theorem:

Theorem 5.1. Assume a compact Riemannian manifold (M, g) without boundary of dimension n ⩾ 2. If at any point x ∈ M the sectional curvature K(e_{i}, e_{j}) is non-positive for any two-direction e_{i} ∧ e_{j} from all the principal orthonormal basis {e_{1}, …, e_{n}} of vectors of the main direction of the Ricci tensor, then any geodesic mapping of (M, g) is affine.

Moreover, we suppose at each point x ∈ M the sectional curvature K(e_{i}, e_{j}) is non-positive and that there is a certain point x_{0} ∈ M where the sectional curvature K(e_{i}, e_{j}) in any two-direction e_{i} ∧ e_{j} of the joint principal orthonormal basis {e_{1}, …, e_{n}} of vectors of the main directions of the Ricci tensor is negative. Then from integral (3.2) follows equation (5.1). On the basis of Theorem 5.1 it follows λ_{1} = ⋯= λ_{n} = 0 and the geodesic mapping is affine.

Further, from integral (5.1a) follows Φ(a) = 0 on M. Then from formula (3.3) at the point x_{0} ∈ M we obtain α_{1} = ⋯= α_{n} = α. Hence a_{ij} = αδ_{ij} , i.e., a_{ij} (x_{0}) = αg_{ij} (x_{0}).

In this case, the affine mapping is homothetic, i.e., ḡ = α′g, where α′ = const. This fact follows from the uniqueness of solutions of the fundamental equations of affine mappings V_{n} → V̄_{n} : ∇_{k} ḡ_{ij} = 0 with initial values ḡ_{ij} (x_{0}) = α′g_{ij} (x_{0}). This is equivalent to a_{ij} (x_{0}) = αg_{ij} (x_{0}), this fact follows from equation (2.3).

We proved the following theorem:

Theorem 5.2. Assume a compact Riemannian manifold (M, g) without boundary of dimension n ⩾ 2. If at any point x ∈ M the sectional curvature K(e_{i}, e_{j}) is non-positive and if there is a certain point x_{0} ∈ M, where the sectional curvature K(e_{i}, e_{j}) is negative in any two-direction e_{i} ∧ e_{j} of all the principal orthonormal basis {e_{1}, …, e_{n}} of vectors of the main directions of the Ricci tensor, then any geodesic mapping of (M, g) is homothetic.

These Theorems generalize the results of Mikeš [15] (see [16]), which were obtained by means of modifications of integral inequalities obtained by Švec [1, p. 10].

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Brno University of Technology Faculty of Civil Engineering Deptartment of Mathematics Brno, Czech Republic

    hinterleitner.irena@seznam.cz