Author | Rovenskii, Vladimir Y. author |
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Title | Foliations on Riemannian Manifolds and Submanifolds [electronic resource] / by Vladimir Y. Rovenskii |
Imprint | Boston, MA : Birkhรคuser Boston, 1998 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4270-3 |
Descript | X, 286 p. online resource |
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