On recurrent spaces of first order
We shall survey our work on Riemannian geometry of tangent sphere bundles with arbitrary constant radius done since the year 2000.
We investigate semi-Riemannian manifolds with pseudosymmetric Weyl curvature tensor satisfying some additional condition imposed on their curvature tensor. Among other things we prove that the so-called Roter type equation holds on such manifolds. We present applications of our results to hypersurfaces in semi-Riemannian space forms, as well as to 4-dimensional warped products.
We investigate hypersurfaces M in spaces of constant curvature with some special minimal polynomial of the second fundamental tensor H of third degree. We present a curvature characterization of pseudosymmetry type for such hypersurfaces. We also prove that if such a hypersurface is a manifold with pseudosymmetric Weyl tensor then it must be pseudosymmetric.
Solutions of the P. J. Ryan problem as well as investigations of curvature properties of Cartan hypersurfaces and Ricci-pseudosymmetric hypersurfaces lead to curvature identities holding on every hypersurface M isometrically immersed in a semi-Riemannian space form. These identities, under some assumptions, give rises to new generalized Einstein metric conditions on M. We investigate hypersurfaces satisfying such curvature conditions.