Foliations on Riemannian Manifolds and Submanifolds

ISBN-13: 9781461287179 / Angielski / Miękka / 2011 / 286 str.

Vladimir Rovenski

Opis

This monograph is based on the author's results on the Riemannian ge ometry of foliations with nonnegative mixed curvature and on the geometry of sub manifolds with generators (rulings) in a Riemannian space of nonnegative curvature. The main idea is that such foliated (sub) manifolds can be decom posed when the dimension of the leaves (generators) is large. The methods of investigation are mostly synthetic. The work is divided into two parts, consisting of seven chapters and three appendices. Appendix A was written jointly with V. Toponogov. Part 1 is devoted to the Riemannian geometry of foliations. In the first few sections of Chapter I we give a survey of the basic results on foliated smooth manifolds (Sections 1.1-1.3), and finish in Section 1.4 with a discussion of the key problem of this work: the role of Riemannian curvature in the study of foliations on manifolds and submanifolds."

Spis treści

I. Foliations on Manifolds.- 1.1 Definitions and examples of foliations.- 1.2 Holonomy.- 1.3 Ehresmann foliations.- 1.4 Foliations and curvature.- II. Local Riemannian Geometry of Foliations.- 2.1 The main tensors and their invariants.- 2.2 A Riemannian almost-product structure.- 2.3 Constructions of geodesic and umbilic foliations.- 2.4 Curvature identities.- 2.5 Riemannian foliations.- III. T-Parallel Fields and Mixed Curvature.- 3.1 Jacobi and Riccati equations.- 3.2 T-parallel vector fields and the Jacobi equation.- 3.3 L-parallel vector fields and variations of curves.- 3.4 Positive mixed curvature.- IV. Rigidity and Splitting of Foliations.- 4.1 Foliations on space forms.- 4.2 Area and volume of a T-parallel vector field.- 4.3 Riccati and Raychaudhuri equations.- V. Submanifolds with Generators.- 5.1 Submanifolds with generators in Riemannian spaces.- 5.2 Submanifolds with generators in space forms.- 5.3 Submanifolds with nonpositive extrinsic q-Ricci curvature..- 5.4 Ruled submanifolds with conditions on mean curvature.- 5.5 Submanifolds with spherical generators.- VI. Decomposition of Ruled Submanifolds.- 6.1 Cylindricity of submanifolds in a Riemannian space of nonnegative curvature.- 6.2 Ruled submanifolds in CROSS and the Segre embedding..- 6.3 Ruled submanifolds in a Riemannian space of positive curvature and Segre type embeddings.- VII. Decomposition of Parabolic Submanifolds.- 7.1 Parabolic submanifolds in CROSS.- 7.2 Parabolic submanifolds in a Riemannian space of positive curvature.- 7.3 Remarks on pseudo-Riemannian isometric immersions.- Appendix A. Great Sphere Foliations and Manifolds with Curvature Bounded Above.- A.1 Great circle foliations.- A.2 Extremal theorem for manifolds with curvature bounded above.- Appendix B. Submersions of Riemannian Manifolds with Compact Leaves.- Appendix C. Foliations by Closed Geodesics with Positive Mixed Sectional Curvature.- References.