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Perspective on Canonical Riemannian Metrics 2020 ed. [Kietas viršelis]

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  • Formatas: Hardback, 247 pages, aukštis x plotis: 235x155 mm, weight: 571 g, 1 Illustrations, color; 1 Illustrations, black and white; XIX, 247 p. 2 illus., 1 illus. in color., 1 Hardback
  • Serija: Progress in Mathematics 336
  • Išleidimo metai: 24-Oct-2020
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 303057184X
  • ISBN-13: 9783030571849
Kitos knygos pagal šią temą:
Perspective on Canonical Riemannian Metrics 2020 ed.Didesnis paveikslėlis
  • Formatas: Hardback, 247 pages, aukštis x plotis: 235x155 mm, weight: 571 g, 1 Illustrations, color; 1 Illustrations, black and white; XIX, 247 p. 2 illus., 1 illus. in color., 1 Hardback
  • Serija: Progress in Mathematics 336
  • Išleidimo metai: 24-Oct-2020
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 303057184X
  • ISBN-13: 9783030571849
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783030571849.html
Raktažodžiai:

This book focuses on a selection of special topics, with emphasis on past and present research of the authors on “canonical” Riemannian metrics on smooth manifolds.

On the backdrop of the fundamental contributions given by many experts in the field, the volume offers a self-contained view of the wide class of “Curvature Conditions” and “Critical Metrics” of suitable Riemannian functionals. The authors describe the classical examples and the relevant generalizations.

This monograph is the winner of the 2020 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics.

Recenzijos

A very readable introduction, which encourages the reader to delve into the subject. The book is carefully written. It is not an encyclopedia of Riemannian geometry, but a deep introduction to active research topics around the idea of searching good metrics on a manifold. Moreover, it offers a good motivation to these topics, showing the relationship among different classes of Riemannian manifolds. (Fernando Etayo Gordejuela, zbMATH 1478.53003, 2022)

Introduction xi
1 Basic Concepts of Riemannian Geometry
1(26)
1.1 Moving frames: Levi-Civita connection, structure equations and curvatures
1(13)
1.1.1 Levi-Civita connection, first structure equation and covariant derivatives
1(5)
1.1.2 Second structure equation and the Riemann tensor
6(4)
1.1.3 Ricci, scalar and sectional curvatures
10(4)
1.2 General frame: everything with Christoffel symbols
14(5)
1.2.1 Levi-Civita connection and covariant derivatives revisited
14(1)
1.2.2 Riemann and Ricci curvatures revisited
15(1)
1.2.3 Covariant derivatives revisited, Hessian and Laplacian
16(3)
1.3 Back to moving frames: the decomposition of the curvature tensor
19(3)
1.4 Other curvatures: A, B, C
22(5)
2 Commutations and Variations
27(24)
2.1 Commutation formulas
27(6)
2.2 Variations of curvatures tensors and other geometric quantities
33(18)
2.2.1 General variations
33(6)
2.2.2 Conformal deformations
39(6)
2.2.3 Aubin's deformation
45(6)
3 The Weyl Tensor
51(20)
3.1 General properties
51(6)
3.1.1 Some formulas
51(2)
3.1.2 Some inequalities
53(4)
3.1.3 The Weyl-Schouten and Aubin theorems
57(1)
3.2 The case of dimension four
57(14)
3.2.1 Self-dual and anti-self-dual parts of W
58(1)
3.2.2 The Derdzinski basis
58(1)
3.2.3 Special formulas
59(6)
3.2.4 Integral identities and topology
65(2)
3.2.5 A Kato inequality
67(4)
4 Curvature Conditions
71(38)
4.1 Old and new canonical metrics: algebraic and analytic conditions
71(6)
4.1.1 Ricci solitons
76(1)
4.2 Canonical metrics revisited: equivalent conditions
77(4)
4.3 The rigid classes: f, f, f and f
81(2)
4.4 The class f
83(6)
4.4.1 Rigidity and characterization results
83(4)
4.4.2 Two examples
87(2)
4.5 The class f: a possible generalization of the Yamabe problem
89(13)
4.5.1 An obstruction
89(2)
4.5.2 An example
91(1)
4.5.3 An existence result in Euclidean space
92(10)
4.6 Non-gradient canonical metrics
102(5)
4.6.1 The class X
104(1)
4.6.2 The classes X and X
104(1)
4.6.3 The class X
105(2)
4.6.4 The class X
107(1)
4.7 Final remarks and open problems
107(2)
5 Critical Metrics of Riemannian Functionals
109(26)
5.1 The Einstein--Hilbert functional
109(3)
5.2 Quadratic curvature functionals
112(3)
5.2.1 A basis
112(1)
5.2.2 Remarks on two special cases
113(2)
5.3 Some rigidity results for quadratic functionals
115(20)
5.3.1 The Euler-Lagrange equations
118(2)
5.3.2 Proofs of Theorem 5.4 and Theorem 5.5
120(6)
5.3.3 Proof of Theorem 5.7
126(2)
5.3.4 Proof of Theorem 5.8
128(3)
5.3.5 The Euler-Lagrange equation for t,s
131(4)
6 Bochner-Weitzenbock Formulas and Applications
135(24)
6.1 A general Bochner-Weitzenbock formula
135(2)
6.1.1 General dimension
135(1)
6.1.2 Dimension four and an integral identity
136(1)
6.2 Some applications
137(8)
6.2.1 Harmonic Weyl and Einstein manifolds: the first Bochner-Weitzenbock formula
137(1)
6.2.2 Rigidity results for Einstein manifolds
138(2)
6.2.3 A general result on four-dimensional manifolds
140(5)
6.3 Higher-order Bochner-Weitzenbock formulas on Einstein manifolds
145(14)
6.3.1 Rough formulas
145(4)
6.3.2 The second Bochner-Weitzenbock formula
149(1)
6.3.3 A rigidity result for Einstein manifolds
150(9)
7 Ricci Solitons: Selected Results
159(54)
7.1 Preliminary results
161(6)
7.1.1 Fundamental formulas for Ricci solitons
161(4)
7.1.2 The tensor D and the integrability conditions
165(2)
7.2 Rigidity I: pointwise conditions
167(7)
7.2.1 Compact shrinkers with strictly positive sectional curvature
167(5)
7.2.2 Further results in the non-necessarily gradient case
172(2)
7.3 Rigidity II: integral conditions
174(17)
7.3.1 The compact case: integral pinching conditions
174(4)
7.3.2 The non-compact case: L1 conditions and integral curvature decay
178(13)
7.4 Rigidity III: vanishing conditions on the Weyl tensor
191(9)
7.4.1 A key integral formula
194(2)
7.4.2 Proof of the results
196(4)
7.5 Rigidity IV: Weyl scalars
200(13)
7.5.1 Weyl scalars on a Ricci soliton
202(3)
7.5.2 Main results
205(3)
7.5.3 Special cases
208(5)
8 Existence Results of Canonical Metrics on Four-Manifolds
213(18)
8.1 A new variational problem: Weak harmonic Weyl metrics
213(4)
8.2 The Euler-Lagrange equation
217(2)
8.2.1 Critical metrics
217(2)
8.2.2 The PDE for the conformal factor
219(1)
8.3 Existence of minimizers
219(12)
8.3.1 Proof of Theorem 8.2
219(1)
8.3.2 Some preliminary results
220(1)
8.3.3 Existence
221(4)
8.3.4 Uniqueness
225(3)
8.3.5 Further results
228(3)
List of Symbols 231(2)
Bibliography 233(10)
Index 243