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.
Auflage
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
Research
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Maße
Höhe: 241 mm
Breite: 160 mm
Dicke: 21 mm
Gewicht
ISBN-13
978-0-8176-3806-1 (9780817638061)
DOI
10.1007/978-1-4612-4270-3
Schweitzer Klassifikation
Vladimir Rovenski (University of Haifa) and Pawe¿ Walczak (University of Lodz), are well-known scientists, specializing in differential geometry, topology and dynamics of foliations. Their scientific contact began in May/June of 1995 during the International Conference "Foliations: Geometry and Dynamics" in Warsaw. Their common interests in Riemannian geometry of foliations and submanifolds sparked the beginning of their scientific co-operation. The authors formed a common theme of research and the idea of a scientific relay race. The scientific relay race was started by Prof. Walczak who had won a Marie Curie grant and conducted research at Institut de Mathématiques de Bourgogne (Dijon, France) from 2003-2005. Prof. Rovenski won a similar Marie Curie grant and conducted research in cooperation with Walczak at the University of Lodz from 2008-2010. Their scientific synergies ongoing, and the scientific relay race is successfully continued by their students. The collaboration and friendship of the authors for over 25 years has led to several scientific works in extrinsic geometry of foliations of Riemannian and Finsler manifolds.
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.
