Extrinsic Geometry of Foliations (Progress in Mathematics, 339)

Description

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. Preface Contents List of Figures 1 Foliations and the Mixed Curvature 1.1 Foliations and Holonomy 1.1.1 Basic Notions 1.1.2 Holonomy 1.1.3 Saturated and Minimal Sets, Generic Leaves 1.2 Metric Structures on Manifolds 1.2.1 Riemannian Structure 1.2.2 Finsler Structure 1.3 Basic of the Extrinsic Geometry of Foliations 1.3.1 Fundamental Tensors 1.3.2 The Mixed Curvature 1.4 The Partial Ricci Curvature 1.4.1 Structures with Constant Partial Ricci Curvature 1.4.2 Prescribing the Partial Ricci Curvature 1.4.3 The Weighted Mixed Curvature 1.4.4 Toponogov Conjecture 1.5 Appendix 1.5.1 Tensors and Differential Forms 1.5.2 Frobenius Theorem 1.5.3 The Elementary Symmetric Functions 2 Integral Formulas 2.1 Codimension One Foliations of Riemannian Manifolds 2.1.1 Using a Family of Diffeomorphisms 2.1.2 Applications 2.1.2.1 Totally Geodesic and Umbilical Foliations 2.1.2.2 Foliations with Conformally Defined Metric 2.1.3 Using the Divergence Theorem 2.2 Foliations and Singularities 2.2.1 Adapted Singular Foliations 2.2.2 Improper Integrals 2.2.3 Civilized Foliations 2.3 Foliations of Arbitrary Codimension 2.3.1 Using a Family of Diffeomorphisms 2.3.1.1 Algebraic Preliminaries 2.3.1.2 The Integral Formulas 2.3.2 Using the Divergence Theorem 2.3.2.1 The Leaf-Wise Divergence 2.3.2.2 The Integral Formulas 2.3.2.3 Totally Geodesic and Umbilical Foliations 2.3.3 Splitting of Weighted Generalized Products 2.3.4 Multi-Product Structures 2.4 Foliations of Metric-Affine Manifolds 2.4.1 Integral Formulas with the Mixed Scalar Curvature 2.4.2 Integral Formula with the Ricci Curvature 2.4.3 Splitting Results 2.4.3.1 Harmonic Distributions 2.4.3.2 Totally Umbilical Distributions 2.5 Codimension One Foliations of Finsler Spaces 2.5.1 The Generalization of (α,β)-Norm 2.5.2 The Modified Scalar Product 2.5.3 The Shape Operator 2.5.4 Around the Reeb Integral Formula and Its Counterpart 3 Prescribing the Mean Curvature 3.1 Minimal Submanifolds 3.2 Tautness of Foliations 3.2.1 Rummler Formula 3.2.2 Foliation Currents 3.2.3 Tautness 3.2.4 Tautness and Holonomy 3.3 Prescribing Mean Curvature in Codimension One 3.3.1 Novikov Components 3.3.2 Consequences of Rummler Formula 3.3.3 Characterization of Mean Curvature Functions 3.4 Prescribing Mean Curvature in Higher Codimension 3.4.1 Notation 3.4.2 Away from Singularities 3.4.3 At Singular Sets 4 Variational Formulae 4.1 ``Optimally Placed'' Distributions 4.2 Adapted Variations of Metric 4.2.1 Variational Formulae 4.2.2 Euler–Lagrange Equations for the Total Smix 4.2.3 Particular Cases 4.2.3.1 Critical Adapted Metrics on Foliations 4.2.3.2 Critical Adapted Metrics on Flows 4.2.3.3 Conformal Submersions 4.3 General Variations of Metric 4.3.1 Variational Formulae 4.3.2 Euler–Lagrange Equations for the Total Smix 4.3.3 Particular Cases 4.3.3.1 Contact Metric Structure 4.3.3.2 Non-integrable Distributions 4.4 Einstein–Hilbert Type Action 4.4.1 Variable Metric 4.4.2 The Mixed Field Equations for Space-Times 4.4.3 Variable Connection 4.5 The Godbillon-Vey Type Invariant 4.5.1 Construction 4.5.2 Variations of (ω,T) and the Index Form 4.5.3 Integrability in Average 4.5.4 Concordance and Homotopy 4.5.5 Critical Foliations 4.5.6 Around the Reinhart–Wood Formula 4.5.7 The Bott Invariant 4.5.8 Higher Dimensional Cases 5 Extrinsic Geometric Flows 5.1 Prescribing the Mean Curvature Vector 5.1.1 D"0365D- and D-Related Geometric Quantities 5.1.2 Existence and Uniqueness 5.1.3 The Codimension-One Case 5.1.4 The Doubly Twisted Products 5.2 Flows of Metrics on Codimension-One Foliations 5.2.1 g-Variations of Mean Curvatures 5.2.2 The Extrinsic Geometric Flow Depending on {fm} 5.2.3 The Generalized Companion Matrix 5.2.4 Searching for Power Sums 5.2.5 Existence and Uniqueness 5.2.6 Extrinsic Geometric Flow on a Foliated Surface 5.3 The Partial Ricci Flow 5.3.1 Preliminaries 5.3.2 Time-Dependent Adapted Metrics 5.3.3 The Leafwise Laplacian of the Curvature Tensor 5.3.4 Toward the Linearization of the Partial Ricci Flow 5.3.5 Evolution of the Curvature Tensor 5.3.6 Evolution of the Extrinsic Geometry 5.3.7 Examples with (Co)Dimension One Foliations 5.3.7.1 Totally Geodesic Foliations of Codimension One 5.3.7.2 Foliations by Geodesics 5.3.8 Around an Almost Contact Structure 5.3.8.1 Weak Almost Contact Structure 5.3.8.2 Weak Almost f-Structure 5.4 Prescribing the Mixed Scalar Curvature 5.4.1 Leafwise Constant Mixed Scalar Curvature 5.4.2 D-Conformal Change of Metric 5.4.3 D-Conformal Flows of Metrics 5.4.4 Prescribing Smix on Warped Products 5.4.5 Prescribing Smix by a D-Conformal Change of Metric 5.5 The Nonlinear Heat Equation 5.5.1 Parabolic PDE's 5.5.2 Stabilization of Solutions of the Nonlinear Heat Equation 5.5.2.1 Comparison ODE 5.5.2.2 Case of Ψ3