Foliations, submanifolds, and mixed curvature. (English) Zbl 0964.53001
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.
Reviewer: Costache Apreutesei (Iaşi)
MSC:
| 53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |
| 57R30 | Foliations in differential topology; geometric theory |
| 53C12 | Foliations (differential geometric aspects) |
| 53C40 | Global submanifolds |
Keywords:
hyperbolic space; foliated manifold; foliations on Riemannian manifold; mixed curvature; splitting foliations; integral formula; totally geodesic foliation; submersion; parabolic submanifold; pseudo-Riemannian isometric immersionReferences:
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