Foliations, submanifolds, and mixed curvature
The main purpose of this survey is to present the differential geometry (d.g.) of foliations with a nonnegative mixed curvature and the d.g. of submanifolds in a Riemannian space of nonnegative curvature. The paper contains: 1) Foliated manifolds; 2) Foliations on Riemannian manifolds (basic facts from the theory of submanifolds, main tensors of a foliation, Riemannian almost-product structure, constructions of totally geodesic and totally umbilical foliations, Riemannian foliations); 3) Foliations and a mixed curvature (foliations and curvature, Jacobi vector fields, foliations by geodesics with \(K_{\text{mix}}> 0\), splitting foliations with nonnegative mixed curvature, integral formulas for mixed curvature, foliations with positive partial mixed curvature, transversal totally geodesic foliations and submersions); 4) Foliations and submanifolds (submanifolds with generators in Riemannian spaces, submanifolds with nonpositive extrinsic \(q\)-dimensional Ricci curvature, submanifolds with generators in Euclidean and Lobachevsky spaces, decomposition of ruled and parabolic submanifolds, pseudo-Riemannian isometric immersions). The methods of study are local and global. Some examples and applications are considered. The exposition is based on the author's results and it is clear.
scientific article; zbMATH DE number 1506092
scientific article