Atnaujinkite slapukų nuostatas

Extrinsic Geometry of Foliations 2021 ed. [Kietas viršelis]

,
  • Formatas: Hardback, 319 pages, aukštis x plotis: 235x155 mm, weight: 670 g, 6 Illustrations, color; 16 Illustrations, black and white; XIII, 319 p. 22 illus., 6 illus. in color., 1 Hardback
  • Serija: Progress in Mathematics 339
  • Išleidimo metai: 23-May-2021
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030700666
  • ISBN-13: 9783030700669
Kitos knygos pagal šią temą:
Extrinsic Geometry of Foliations 2021 ed.Didesnis paveikslėlis
  • Formatas: Hardback, 319 pages, aukštis x plotis: 235x155 mm, weight: 670 g, 6 Illustrations, color; 16 Illustrations, black and white; XIII, 319 p. 22 illus., 6 illus. in color., 1 Hardback
  • Serija: Progress in Mathematics 339
  • Išleidimo metai: 23-May-2021
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030700666
  • ISBN-13: 9783030700669
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783030700669.html
Raktažodžiai:

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.


Recenzijos

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 Cisma, zbMATH 1479.53002, 2022)

1 Foliations And The Mixed Curvature
1(60)
1.1 Foliations and Holonomy
1(11)
1.1.1 Basic Notions
2(2)
1.1.2 Holonomy
4(5)
1.1.3 Saturated and Minimal Sets, Generic Leaves
9(3)
1.2 Metric Structures on Manifolds
12(11)
1.2.1 Riemannian Structure
13(7)
1.2.2 Finsler Structure
20(3)
1.3 Basic of the Extrinsic Geometry of Foliations
23(9)
1.3.1 Fundamental Tensors
24(3)
1.3.2 The Mixed Curvature
27(5)
1.4 The Partial Ricci Curvature
32(16)
1.4.1 Structures with Constant Partial Ricci Curvature
32(4)
1.4.2 Prescribing the Partial Ricci Curvature
36(4)
1.4.3 The Weighted Mixed Curvature
40(3)
1.4.4 Toponogov Conjecture
43(5)
1.5 Appendix
48(13)
1.5.1 Tensors and Differential Forms
48(6)
1.5.2 Frobenius Theorem
54(3)
1.5.3 The Elementary Symmetric Functions
57(4)
2 Integral Formulas
61(62)
2.1 Codimension One Foliations of Riemannian Manifolds
61(10)
2.1.1 Using a Family of Diffeomorphisms
63(2)
2.1.2 Applications
65(2)
2.1.3 Using the Divergence Theorem
67(4)
2.2 Foliations and Singularities
71(8)
2.2.1 Adapted Singular Foliations
72(2)
2.2.2 Improper Integrals
74(2)
2.2.3 Civilized Foliations
76(3)
2.3 Foliations of Arbitrary Codimension
79(17)
2.3.1 Using a Family of Diffeomorphisms
79(5)
2.3.2 Using the Divergence Theorem
84(6)
2.3.3 Splitting of Weighted Generalized Products
90(1)
2.3.4 Multi-Product Structures
91(5)
2.4 Foliations of Metric-Affine Manifolds
96(7)
2.4.1 Integral Formulas with the Mixed Scalar Curvature
96(3)
2.4.2 Integral Formula with the Ricci Curvature
99(2)
2.4.3 Splitting Results
101(2)
2.5 Codimension One Foliations of Finsler Spaces
103(20)
2.5.1 The Generalization of (a, 0)-Norm
104(4)
2.5.2 The Modified Scalar Product
108(6)
2.5.3 The Shape Operator
114(4)
2.5.4 Around the Reeb Integral Formula and Its Counterpart
118(5)
3 Prescribing The Mean Curvature
123(30)
3.1 Minimal Submanifolds
123(1)
3.2 Tautness of Foliations
124(9)
3.2.1 Rummler Formula
124(2)
3.2.2 Foliation Currents
126(2)
3.2.3 Tautness
128(2)
3.2.4 Tautness and Holonomy
130(3)
3.3 Prescribing Mean Curvature in Codimension One
133(7)
3.3.1 Novikov Components
134(2)
3.3.2 Consequences of Rummler Formula
136(1)
3.3.3 Characterization of Mean Curvature Functions
136(4)
3.4 Prescribing Mean Curvature in Higher Codimension
140(13)
3.4.1 Notation
141(2)
3.4.2 Away from Singularities
143(5)
3.4.3 At Singular Sets
148(5)
4 Variational Formulae
153(70)
4.1 "Optimally Placed" Distributions
153(3)
4.2 Adapted Variations of Metric
156(19)
4.2.1 Variational Formulae
157(4)
4.2.2 Euler-Lagrange Equations for the Total Smix
161(6)
4.2.3 Particular Cases
167(8)
4.3 General Variations of Metric
175(13)
4.3.1 Variational Formulae
176(2)
4.3.2 Euler-Lagrange Equations for the Total Smix
178(3)
4.3.3 Particular Cases
181(7)
4.4 Einstein-Hilbert Type Action
188(10)
4.4.1 Variable Metric
189(3)
4.4.2 The Mixed Field Equations for Space-Times
192(2)
4.4.3 Variable Connection
194(4)
4.5 The Godbillon-Vey Type Invariant
198(25)
4.5.1 Construction
200(2)
4.5.2 Variations of (co, T) and the Index Form
202(4)
4.5.3 Integrability in Average
206(3)
4.5.4 Concordance and Homotopy
209(2)
4.5.5 Critical Foliations
211(4)
4.5.6 Around the Reinhart-Wood Formula
215(4)
4.5.7 The Bott Invariant
219(1)
4.5.8 Higher Dimensional Cases
220(3)
5 Extrinsic Geometric Flows
223(80)
5.1 Prescribing the Mean Curvature Vector
223(12)
5.1.1 D- and D-Related Geometric Quantities
225(3)
5.1.2 Existence and Uniqueness
228(5)
5.1.3 The Codimension-One Case
233(2)
5.1.4 The Doubly Twisted Products
235(1)
5.2 Flows of Metrics on Codimension-One Foliations
235(16)
5.2.1 gT-Variations of Mean Curvatures
237(1)
5.2.2 The Extrinsic Geometric Flow Depending on {fm}
238(2)
5.2.3 The Generalized Companion Matrix
240(2)
5.2.4 Searching for Power Sums
242(2)
5.2.5 Existence and Uniqueness
244(5)
5.2.6 Extrinsic Geometric Flow on a Foliated Surface
249(2)
5.3 The Partial Ricci Flow
251(21)
5.3.1 Preliminaries
251(2)
5.3.2 Time-Dependent Adapted Metrics
253(2)
5.3.3 The Leafwise Laplacian of the Curvature Tensor
255(3)
5.3.4 Toward the Linearization of the Partial Ricci Flow
258(2)
5.3.5 Evolution of the Curvature Tensor
260(3)
5.3.6 Evolution of the Extrinsic Geometry
263(1)
5.3.7 Examples with (Co)Dimension One Foliations
264(4)
5.3.8 Around an Almost Contact Structure
268(4)
5.4 Prescribing the Mixed Scalar Curvature
272(16)
5.4.1 Leafwise Constant Mixed Scalar Curvature
273(2)
5.4.2 D-Conformal Change of Metric
275(2)
5.4.3 D-Conformal Flows of Metrics
277(2)
5.4.4 Prescribing Smtx on Warped Products
279(6)
5.4.5 Prescribing Smix by a D-Conformal Change of Metric
285(3)
5.5 The Nonlinear Heat Equation
288(15)
5.5.1 Parabolic PDE's
288(4)
5.5.2 Stabilization of Solutions of the Nonlinear Heat Equation
292(11)
References 303(10)
Index 313(6)
About the Authors 319
Vladimir Rovenski  (University of Haifa) and Pawe 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 20032005. Prof. Rovenski won a similar Marie Curie grant and conducted research in cooperation with Walczak at the University of Lodz from 20082010. 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.