Extrinsic Geometry of Foliations

1. Front Matter

   Pages i-xiii

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2. Foliations and the Mixed Curvature

   • Vladimir Rovenski, Paweł Walczak

   Pages 1-59

3. Integral Formulas

   • Vladimir Rovenski, Paweł Walczak

   Pages 61-122

4. Prescribing the Mean Curvature

   • Vladimir Rovenski, Paweł Walczak

   Pages 123-151

5. Variational Formulae

   • Vladimir Rovenski, Paweł Walczak

   Pages 153-221

6. Extrinsic Geometric Flows

   • Vladimir Rovenski, Paweł Walczak

   Pages 223-302

7. Back Matter

   Pages 303-319

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This book is devoted to geometric problems of foliation theory, in particular those related to extrinsic geometry, modern branch of Riemannian Geometry. The concept of mixed curvature is central to the discussion, and a version of the deep problem of the Ricci curvature for the case of mixed curvature of foliations is examined. The book is divided into five chapters that deal with integral and variation formulas and curvature and dynamics of foliations. Different approaches and methods (local and global, regular and singular) in solving the problems are described using integral and variation formulas, extrinsic geometric flows, generalizations of the Ricci and scalar curvatures, pseudo-Riemannian and metric-affine geometries, and 'computable' Finsler metrics.

The book presents the state of the art in geometric and analytical theory of foliations as a continuation of the authors' life-long work in extrinsic geometry.  It is designed for newcomers to the field as well as experienced geometers working in Riemannian geometry, foliation theory, differential topology, and a wide range of researchers in differential equations and their applications.  It may also be a useful supplement to postgraduate level work and can inspire new interesting topics to explore.

• foliations
• extrinsic geometry
• Ricci flow
• curvature
• integral formulas
• variation formulas
• mean curvature
• Yamabe problem
• Toponogov's conjecture
• MInkowski norms
• co-dimension one foliations
• Einstein-Hilbert
• Godbillon-Vey

“The reader is assumed to have some background in topology and differential geometry. The book is a continuation of the authors’ work in extrinsic geometry and thus provides a useful reference for researchers in this field.” (Emanuel-Ciprian Cismaş, zbMATH 1479.53002, 2022)

• Department of Mathematics, University of Haifa, Haifa, Israel

  Vladimir Rovenski

• Faculty of Mathematics and Computer Science, University of Lodz, Łódź, Poland

  Paweł Walczak

Vladimir Rovenski  (University of Haifa) and Paweł 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.

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