Foliations on Riemannian Manifolds and SubmanifoldsThis 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. |
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B₂ codim codimension compact Riemannian complete condition constant curvature Corollary curvature tensor curve cylinder defined Definition Diff dimension Ehresmann connection Euclidean space exists fibers finite formula function Geom geometry h₁ holonomy Hopf fibration hypersurfaces inequality integrable isometric immersion Jacobi vector field k₁ Kählerian Kmix L-parallel leaf leaves Lemma linear M₁ Math matrix mean curvature nonnegative normal bundle normal vector obtain orthogonal distribution orthonormal parabolic submanifolds proof of Theorem relative nullity Riccati equation Ricci curvature Riemannian foliations Riemannian manifold Riemannian space Riemannian submersions ruled submanifold Sasaki metric satisfying second fundamental form sectional curvature space form sphere strongly parabolic structural tensor subspace symmetric T₁ T₂ tangent TL¹ topology totally geodesic foliation totally geodesic submanifold totally umbilic transversal unit vector V₁ y₁