Topics in Extrinsic Geometry of Codimension - One Foliations

by Rovenski, Vladimir; Walczak, Pawel

• ISBN13: 9781441999078
• ISBN10: 1441999078
• Format: Paperback
• Copyright: 2011-07-26
• Publisher: Springer Verlag

Summary

Extrinsic geometry describes those properties of foliations on Riemannian manifolds which can be expressed in terms of the second fundamental form of the leaves.The Integral Formulae, introduced in chapter 1, are useful for problems such as: prescribing higher mean curvatures of foliations, minimizing volume and energy defined for vector or plane fields on manifolds, and existence of foliations whose leaves enjoy given geometric properties.The central topic of this book is Extrinsic Geometric Flow (EGF) on foliated manifolds. EGF may be a tool for prescribing extrinsic geometric properties of foliations.To develop EGF, one needs Variation Formulae, discovered in chapter 2, which express variation of different quantities belonging to extrinsic geometry of a fixed foliation under variation of the Riemannian Structure of the ambient manifold.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.