Global calculus / S. Ramanan.

Format:

Book

Author/Creator:

Ramanan, S.

Series:

Graduate studies in mathematics 1065-7339 ; v. 65.

Graduate studies in mathematics, 1065-7339 ; v. 65

Language:

English

Subjects (All):

Geometry, Algebraic.

Differential operators.

Analytic spaces.

Geometry, Differential.

Physical Description:

xi, 316 pages : illustrations ; 26 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society, 2005.

Contents:

Chapter 1 Sheaves and Differential Manifolds: Definitions and Examples 1

1 Sheaves and Presheaves 2

2 Basic Constructions 9

3 Differential Manifolds 12

4 Lie Groups; Action on a Manifold 23

Chapter 2 Differential Operators 27

1 First Order Differential Operators 27

2 Locally Free Sheaves and Vector Bundles 29

3 Flow of a Vector Field 38

4 Theorem of Frobenius 46

5 Tensor Fields; Lie Derivative 50

6 The Exterior Derivative; de Rham Complex 54

7 Differential Operators of Higher Order 61

Chapter 3 Integration on Differential Manifolds 73

1 Integration on a Manifold 73

2 Sheaf of Densities 79

3 Adjoints of Differential Operators 85

Chapter 4 Cohomology of Sheaves and Applications 93

1 Injective Sheaves 93

2 Sheaf Cohomology 98

3 Cohomology through Other Resolutions 105

4 Singular and Sheaf Cohomologies 107

5 Cech and Sheaf Cohomologies 114

6 Differentiable Simplices; de Rham's Theorem 117

Chapter 5 Connections on Principal and Vector Bundles; Lifting of Symbols 125

1 Connections in a Vector Bundle 125

2 The Space of All Connections on a Bundle 131

3 Principal Bundles 135

4 Connections on Principal Bundles 143

5 Curvature 148

6 Chern-Weil Theory 154

7 Holonomy Group; Ambrose-Singer Theorem 164

Chapter 6 Linear Connections 171

1 Linear Connections 171

2 Lifting of Symbols and Torsion 177

Chapter 7 Manifolds with Additional Structures 185

1 Reduction of the Structure Group 185

2 Torsion Free G-Connections 194

3 Complex Manifolds 197

4 The Outer Gauge Group 201

5 Riemannian Geometry 207

6 Riemannian Curvature Tensor 211

7 Ricci, Scalar and Weyl Curvature Tensors 219

8 Clifford Structures and the Dirac Operator 224

Chapter 8 Local Analysis of Elliptic Operators 229

1 Regularisation 229

2 A Characterisation of Densities 232

3 Schwartz Space of Functions and Densities 233

4 Fourier Transforms 238

5 Distributions 242

6 Theorem of Sobolev 245

7 Interior Regularity of Elliptic Solutions 251

Chapter 9 Vanishing Theorems and Applications 257

1 Elliptic Operators on Differential Manifolds 257

2 Elliptic Complexes 261

3 Composition Formula 270

4 A Vanishing Theorem 276

5 Hodge Decomposition 279

6 Lefschetz Decomposition 281

7 Kodaira's Vanishing Theorem 287

8 The Imbedding Theorem 292

1 Algebra 301

2 Topology 305

3 Analysis 307.

Notes:

Includes bibliographical references (pages 311-312) and index.

ISBN:

0821837028

OCLC:

56192379