**Graduate Studies in Mathematics**

Volume: 71;
2006;
633 pp;
Hardcover

MSC: Primary 53;
Secondary 57

Print ISBN: 978-0-8218-3929-4

Product Code: GSM/71

List Price: $86.00

Individual Member Price: $68.80

**Electronic ISBN: 978-1-4704-2110-6
Product Code: GSM/71.E**

List Price: $86.00

Individual Member Price: $68.80

#### Supplemental Materials

# Modern Geometric Structures and Fields

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*S. P. Novikov; I. A. Taimanov*

The book presents the basics of Riemannian geometry in its modern form as geometry of differentiable manifolds and the most important structures on them. The authors' approach is that the source of all constructions in Riemannian geometry is a manifold that allows one to compute scalar products of tangent vectors. With this approach, the authors show that Riemannian geometry has a great influence to several fundamental areas of modern mathematics and its applications. In particular,

- Geometry is a bridge between pure mathematics and natural sciences, first of all physics. Fundamental laws of nature are formulated as relations between geometric fields describing various physical quantities.
- The study of global properties of geometric objects leads to the far-reaching development of topology, including topology and geometry of fiber bundles.
- Geometric theory of Hamiltonian systems, which describe many physical phenomena, led to the development of symplectic and Poisson geometry. Field theory and the multidimensional calculus of variations, presented in the book, unify mathematics with theoretical physics.
- Geometry of complex and algebraic manifolds unifies Riemannian geometry with modern complex analysis, as well as with algebra and number theory.

Prerequisites for using the book include several basic undergraduate courses, such as advanced calculus, linear algebra, ordinary differential equations, and elements of topology.

#### Table of Contents

# Table of Contents

## Modern Geometric Structures and Fields

Table of Contents pages: 1 2

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface to the English Edition xiii14 free
- Preface xvii18 free
- Chapter 1. Cartesian Spacesand Euclidean Geometry 122 free
- Chapter 2. Symplectic and Pseudo-Euclidean Spaces 3556
- Chapter 3. Geometry of Two-Dimensional Manifolds 5374
- Chapter 4. Complex Analysis in the Theory of Surfaces 85106
- 4.1. Complex spaces and analytic functions 85106
- 4.2. Geometry of the sphere 94115
- 4.3. Geometry of the pseudosphere 100121
- 4.4. The theory of surfaces in terms of a conformal parameter 107128
- 4.4.1. Existence of a conformal parameter 107128
- 4.4.2. The basic equations in terms of a conformal parameter 110131
- 4.4.3. Hopf differential and its applications 112133
- 4.4.4. Surfaces of constant Gaussian curvature. The Liouvilleequation 113134
- 4.4.5. Surfaces of constant mean curvature. The sinh-Gordonequation 115136

- 4.5. Minimal surfaces 117138
- Exercises to Chapter 4 122143

- Chapter 5. Smooth Manifolds 125146
- 5.1. Smooth manifolds 125146
- 5.1.1. Topological and metric spaces 125146
- 5.1.2. On the notion of smooth manifold 129150
- 5.1.3. Smooth mappings and tangent spaces 133154
- 5.1.4. Multidimensional surfaces in R[sup(n)]. Manifolds withboundary 137158
- 5.1.5. Partition of unity. Manifolds as multidimensionalsurfaces in Euclidean spaces 141162
- 5.1.6. Discrete actions and quotient manifolds 143164
- 5.1.7. Complex manifolds 145166

- 5.2. Groups of transformations as manifolds 156177
- 5.3. Quaternions and groups of motions 170191
- Exercises to Chapter 5 175196

- Chapter 6. Groups of Motions 177198
- 6.1. Lie groups and algebras 177198
- 6.1.1. Lie groups 177198
- 6.1.2. Lie algebras 179200
- 6.1.3. Main matrix groups and Lie algebras 187208
- 6.1.4. Invariant metrics on Lie groups 193214
- 6.1.5. Homogeneous spaces 197218
- 6.1.6. Complex Lie groups 204225
- 6.1.7. Classification of Lie algebras 206227
- 6.1.8. Two-dimensional and three-dimensional Lie algebras 209230
- 6.1.9. Poisson structures 212233
- 6.1.10. Graded algebras and Lie super algebras 217238

- 6.2. Crystallographic groups and their generalizations 221242
- Exercises to Chapter 6 242263

- Chapter 7. Tensor Algebra 245266
- Chapter 8. Tensor Fields in Analysis 285306
- Chapter 9. Analysis of Differential Forms 315336
- Chapter 10. Connections and Curvature 351372
- 10.1. Covariant differentiation 351372
- 10.2. Curvature tensor 369390
- 10.2.1. Definition of the curvature tensor 369390
- 10.2.2. Symmetries of the curvature tensor 372393
- 10.2.3. The Riemann tensors in small dimensions, the Riccitensor, scalar and sectional curvatures 374395
- 10.2.4. Tensor of conformal curvature 377398
- 10.2.5. Tetrad formalism 380401
- 10.2.6. The curvature of invariant metrics of Lie groups 381402

- 10.3. Geodesic lines 383404
- Exercises to Chapter 10 392413

- Chapter 11. Conformal and Complex Geometries 397418

Table of Contents pages: 1 2

#### Readership

Graduate students and research mathematicians interested in modern geometry and its applications.

#### Reviews

The book is designed for students in mathematics and theoretical physics but it will be very useful for teachers as well. ...has a much wider scope than the usual textbook on differential geometry.

-- European Mathematical Society Newsletter

The textbook offers an abundance of general theories, concrete examples, and algebraic computations. It is a readable introduction to a wide number of areas of geometrical and algebraic themes in its interrelation with physics, appropriate for students of mathematics and theoretical physics.

-- Hubert Gollek for Zentralblatt MATH

This excellent textbook offers a modern treatment of most differential geometrical notions and tools used today, in pure mathematics as well as in theoretical physics. The approach used by the authors is most remarkable...the reviewer thinks that this is an outstanding book, highly recommended to mathematicians and mathematical physicists, from beginners up to advanced researchers.

-- Charles-Michel Marle for Mathematical Reviews