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Geodesic mappings of Riemannian spaces. (Geodezicheskie otobrazheniya rimanovykh prostranstv). (Russian) Zbl 0637.53020
Moskva: Nauka. 256 p. R. 1.60 (1979).
This pleasant book will be welcomed by those who read a little Russian and who have a nostalgia for differential geometry in the classical manner, done locally with tensor analysis and geometric objects, without exterior forms, fibre bundles, or any global worries. A geodesic mapping here is an invertible differentiable mapping of one n-dimensional affinely connected or Riemannian manifold onto another, such that geodesics are mapped into geodesics, the affine parameter not in general being preserved; if the mapping sends \(x^ 1,...,x^ n\) to \(\bar x{}^ 1,...,\bar x^ n\), then the necessary and sufficient condition is: \({\bar \Gamma}^ h_{ij}(x)=\Gamma^ h_{ij}(x)+\psi_ i(x)\delta_ j^ h+\psi_ j(x)\delta_ i^ h,\) for some vector \(\phi_ i(x).\)
The book has five chapters, the first being a self-contained summary of all the definitions and basic results needed later. The second chapter covers the classical results due to Levi-Civita, T. Y. Thomas and others, plus some of the author’s results along the same lines. In the third chapter the author presents his own methods for deciding if a given Riemannian manifold admits nontrivial geodesic mappings, and for finding all the possible target manifolds in case it does. The fourth chapter gives the author’s results on almost geodesic mappings - a generalized concept which concerns some special types of geometries and mappings studied by various authors (mostly the theory of \((n-2)\)-projective spaces due to Vrănceanu, P. A. Shirokov, and Kagan). This work continues in the last chapter for the case of holomorphically projective mappings of Kähler manifolds (Ōtsuki, Tashiro); the results here are mostly those of the author, V. V. Domashev, and J. Mikesh.

MSC:
53B20 Local Riemannian geometry
53B35 Local differential geometry of Hermitian and Kählerian structures
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry