Topics in extrinsic geometry of codimension-one foliations

Extrinsic geometry describes properties of foliations on Riemannian manifoldswhich can be expressed in terms of the second fundamental form of the leaves.The authors of Topics in Extrinsic Geometry of Codimension-One Foliations achieve a technical tour de force, which will lead to important geometric results. The Integral Formulae , introduced in chapter 1, is a useful for problems such as: prescribing higher mean curvatures of foliations, minimizing volume andenergy defined for vector or plane fields on manifolds, and existence of foliations whose leaves enjoy given geometric properties. The Integral Formulae steams from a Reeb formula, for foliations on space forms which generalize the classical ones. For a special auxiliary functions the formulae involve the Newton transformations of the Weingarten operator. The central topic of this book is Extrinsic Geometric Flow (EGF) on foliated manifolds, which may be a tool for prescribing extrinsic geometric properties of foliations. To develop EGF, one needs Variational Formulae , revealed in chapter 2, which expresses a change in different extrinsic geometric quantities of a fixed foliation under leaf-wise variation of the Riemannian Structure of the ambient manifold. Chapter 3 defines a general notion of EGF and studies the evolution of Riemannian metrics along the trajectories of this flow(e.g., describes the short-time existenceand uniqueness theory and estimate the maximal existence time).Some special solutions (called Extrinsic Geometric Solutions ) of EGF are presented and are of great interest, since they provide Riemannian Structures with very particular geometry of the leaves. This work is aimed at those who have an interest inthe differential geometry of submanifolds and foliations of Riemannian manifolds. New topic of 'foliation with a time-dependent metric' is developed. Presents new research tools in geometry of foliations (Extrinsic Geometric Flow). Presents examples and open problems for foliated surfaces. INDICE: -1. Integral Formulae (Introduction, Preliminaries, Integral Formulae for Codimension-one foliations).-2.Variation Formulae (Introduction, Auxiliary results, Variations of extrinsic geometric quantities, Variations of general functional, Variations of particular functional, Applications and examples).-3. Extrinsic Geometric Flows (Introduction, The systems of PDE’s related toEGF, Auxiliary results, Existence and uniqueness results, A solution to general case, Global existence of EGF, Variation formulae for EGF, Extrinsic geometric solitons, Applications and examples). References.

• ISBN: 978-1-4419-9907-8
• Editorial: Springer New York
• Encuadernacion: Rústica
• Páginas: 96
• Fecha Publicación: 06/08/2011
• Nº Volúmenes: 1
• Idioma: Inglés