Foliations on Riemannian Manifolds and Submanifolds

The ideas and methods of foliations are very popular in mathematics and its applications. The key problem of this book is the role of a Riemannian curvature in studies of manifolds and submanifolds with foliations. Rovenskii discusses the results of many geometers, but the book principally focuses on the author's own investigations into the Riemannian geometry of foliations and submanifolds with generators having nonnegative curvature. The main idea is that such manifolds are decomposed into a direct product when the dimension of leaves is sufficiently large. Part I starts with a short introduction (Chapter 1) to the geometry of foliations and continues with local and global results on Riemannian manifolds with foliations, including rigidity, splitting, and integral formulas (Chapter 2-4). In Part 2 Rovenskii gives a survey of submanifolds with generators (Chapter 5) and then combines variational methods (developed in Part I) with synthetic procedures to obtain rigidity and Segre type decomposition of such submanifolds (Chapters 6 & 7). Appendix A, written jointly with V. Toponogov, contains facts on geodesic foliations of round sphere in relation to manifolds of positive sectional curvature bounded from above. The main result generalizes the minimal diameter theorem by M. Berger. This book is intended for students and researchers with a basic knowledge of differential and Riemannian geometry.