Foliations on Riemannian Manifolds and Submanifolds

AvVladimir Rovenski

Inbunden, Engelska, 1997

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Beskrivning

This text's key issue is the role of a Riemannian curvature in studies of manifolds and submanifolds with foliations. The results of many geometers are discussed, but this book principally focuses on the Riemannian geometry of manifolds and submanifolds with generators that have non-negative curvature, the main idea being that such manifolds decompose into a direct product when the dimension of leaves is sufficiently large.

Produktinformation

Utgivningsdatum:1997-12-29
Mått:155 x 235 x 21 mm
Vikt:617 g
Format:Inbunden
Språk:Engelska
Antal sidor:286
Upplaga:1998
Förlag:Birkhauser Boston
ISBN:9780817638061

Utforska kategorier

Matematik inom Naturvetenskap och teknik

Innehållsförteckning

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.