Geodesic mappings and deformations of Riemannian spaces. (Q2710090)

The first five chapters of this 130-page monograph are devoted to a summary of the theory of geodesics and geodesic mappings of Riemannian spaces. They include important historical data concerning the formation of the theory; the persons who have brought in the greatest contribution to the theory are mentioned and the basic results are given with proofs. A diffeomorphism of two Riemannian spaces is called a geodesic transformation if the image of every geodesic is a geodesic. Such a diffeomorphism is nontrivial if it is not affine. The main question of the theory -- on a common and effective criterion for the existence of a nontrivial geodesic transformation -- is hitherto unsolved. In the monograph the well-known classes of Riemannian spaces that admit and do not admit such transformation are listed. The qualitative result received in 1966 reduces the main question to the decision of a Cauchy-type system of differential equations. This result, for example, has allowed finding out lacunas in degrees of the Riemannian space mobility. NEWLINENEWLINENEWLINEIn Chapters 6-8 results of the authors are stated. The basic idea consists in considering infinitesimal geodesic deformations, i.e. infinitesimal transformations that preserve geodesic lines. It is proved, that a Riemannian manifold admits such a transformation if and only if it admits an infinitesimal geodesic deformation. The existence of the last is reduced to the decision of a Cauchy-type system as well. In the conclusion the authors construct an interesting example of a nontrivial global geodesic deformation of compact surfaces of revolution. NEWLINENEWLINENEWLINEThe list of references contains more than 250 entries.