Topics in Extrinsic Geometry of Codimension-One Foliations

Extrinsic geometry describes properties of foliations on Riemannian manifolds which 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 and energy 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 existence and 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 in the differential geometry of submanifolds and foliations of Riemannian manifolds.   

• foliation of codimension one
• geometric flow
• integral formula
• mean curvatures
• variation formula
• partial differential equations

From the reviews:

“There are three chapters in this research monograph, each devoted to a different aspect of the extrinsic geometry of Ƒ. … This book generalizes well-known results but also covers new ground. It is rich in ideas for those who are interested in the geometry of codimension-one foliations.” (James Hebda, Zentralblatt MATH, Vol. 1228, 2012)

“The aim of this research monograph is to study several topics in extrinsic geometry of codimension-one foliations, i.e., topics related to properties of foliations which can be expressed in terms of the second fundamental form of the leaves and its invariants … . The book is very well written and contains a lot of results that will be interesting to specialists and also to differential and Riemannian geometers who are not necessarily experts in the field of foliations.” (Paolo Mastrolia, Mathematical Reviews, Issue 2012 m)

• , Department of Mathematics & Computer Sci, University of Haifa, Haifa, Israel

  Vladimir Rovenski

• , Department of Mathematics, University of Lodz, Lodz, Poland

  Paweł Walczak

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