Global calculus / S. Ramanan.

Format:

Book

Author/Creator:

Ramanan, S., author.

Series:

Graduate studies in mathematics ; Volume 65.

Graduate Studies in Mathematics, 1065-7339 ; Volume 65

Language:

English

Subjects (All):

Geometry, Algebraic.

Differential operators.

Analytic spaces.

Geometry, Differential.

Physical Description:

1 online resource (330 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2005.

Language Note:

English

Summary:

The power that analysis, topology and algebra bring to geometry has revolutionized the way geometers and physicists look at conceptual problems. Some of the key ingredients in this interplay are sheaves, cohomology, Lie groups, connections and differential operators. In Global Calculus, the appropriate formalism for these topics is laid out with numerous examples and applications by one of the experts in differential and algebraic geometry. Ramanan has chosen an uncommon but natural path through the subject. In this almost completely self-contained account, these topics are developed from scratch. The basics of Fourier transforms, Sobolev theory and interior regularity are proved at the same time as symbol calculus, culminating in beautiful results in global analysis, real and complex. Many new perspectives on traditional and modern questions of differential analysis and geometry are the hallmarks of the book. The book is suitable for a first year graduate course on global analysis.

Contents:

""Cover""; ""Title""; ""Copyright""; ""Contents""; ""Preface""; ""Chapter 1. Sheaves and Differential Manifolds: Definitions and Examples""; ""1. Sheaves and Presheaves""; ""2. Basic Constructions""; ""3. Differential Manifolds""; ""4. Lie Groups; Action on a Manifold""; ""Exercises""; ""Chapter 2. Differential Operators""; ""1. First Order Differential Operators""; ""2. Locally Free Sheaves and Vector Bundles""; ""3. Flow of a Vector Field""; ""4. Theorem of Frobenius""; ""5. Tensor Fields; Lie Derivative""; ""6. The Exterior Derivative; de Rham Complex""

""7. Differential Operators of Higher Order""""Exercises""; ""Chapter 3. Integration on Differential Manifolds""; ""1. Integration on a Manifold""; ""2. Sheaf of Densities""; ""3. Adjoints of Differential Operators""; ""Exercises""; ""Chapter 4. Cohomology of Sheaves and Applications""; ""1. Injective Sheaves""; ""2. Sheaf Cohomology""; ""3. Cohomology through Other Resolutions""; ""4. Singular and Sheaf Cohomologies""; ""5. Cech and Sheaf Cohomologies""; ""6. Differentiable Simplices; de Rham's Theorem""; ""Exercises""; ""Chapter 5. Connections on Principal and Vector Bundles

Lifting of Symbols""""1. Connections in a Vector Bundle""; ""2. The Space of All Connections on a Bundle""; ""3. Principal Bundles""; ""4. Connections on Principal Bundles""; ""5. Curvature""; ""6. Chern-Weil Theory""; ""7. Holonomy Group; Ambrose-Singer Theorem""; ""Exercises""; ""Chapter 6. Linear Connections""; ""1. Linear Connections""; ""2. Lifting of Symbols and Torsion""; ""Exercises""; ""Chapter 7. Manifolds with Additional Structures""; ""1. Reduction of the Structure Group""; ""2. Torsion Free G-Connections""; ""3. Complex Manifolds""; ""4. The Outer Gauge Group""

""5. Riemannian Geometry""""6. Riemannian Curvature Tensor""; ""7. Ricci, Scalar and Weyl Curvature Tensors""; ""8. Clifford Structures and the Dirac Operator""; ""Chapter 8. Local Analysis of Elliptic Operators""; ""1. Regularisation""; ""2. A Characterisation of Densities""; ""3. Schwartz Space of Functions and Densities""; ""4. Fourier Transforms""; ""5. Distributions""; ""6. Theorem of Sobolev""; ""7. Interior Regularity of Elliptic Solutions""; ""Chapter 9. Vanishing Theorems and Applications""; ""1. Elliptic Operators on Differential Manifolds""; ""2. Elliptic Complexes""

""3. Composition Formula""""4. A Vanishing Theorem""; ""5. Hodge Decomposition""; ""6. Lefschetz Decomposition""; ""7. Kodaira's Vanishing Theorem""; ""8. The Imbedding Theorem""; ""Appendix""; ""1. Algebra""; ""2. Topology""; ""3. Analysis""; ""Bibliography""; ""Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""K""; ""L""; ""M""; ""N""; ""O""; ""P""; ""R""; ""S""; ""T""; ""V""; ""W""; ""Back Cover""

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-2107-0