Modern geometric structures and fields / S.P. Novikov, I.A. Taimanov ; translated by Dmitry Chibisov.

Format:

Book

Author/Creator:

Novikov, S. P. (Sergeĭ Petrovich)

Taĭmanov, I. A. (Iskander Asanovich), 1961- author.

Contributor:

Chibisov, D. M. (Dmitriĭ Mikhaĭlovich), translator.

Series:

Graduate studies in mathematics ; v. 71.

Graduate studies in mathematics, 1065-7339 ; volume 71

Standardized Title:

Sovremennye geometricheskie struktury i poli͡a. English

Language:

English

Subjects (All):

Geometry, Riemannian.

Differentiable manifolds.

Physical Description:

1 online resource (658 p.)

Edition:

[English edition].

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2006]

Language Note:

English

Summary:

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.

Contents:

""Cover""; ""Title""; ""Copyright""; ""Contents""; ""Preface to the English Edition""; ""Preface""; ""Chapter 1. Cartesian Spacesand Euclidean Geometry""; ""1.1. Coordinates. Space-time""; ""1.1.1. Cartesian coordinates""; ""1.1.2. Change of coordinates""; ""1.2. Euclidean geometry and linear algebra""; ""1.2.1. Vector spaces and scalar products""; ""1.2.2. The length of a curve""; ""1.3. Affine transformations""; ""1.3.1. Matrix formalism. Orientation""; ""1.3.2. Affine group""; ""1.3.3. Motions of Euclidean spaces""; ""1.4. Curves in Euclidean space""

""1.4.1. The natural parameter and curvature""""1.4.2. Curves on the plane""; ""1.4.3. Curvature and torsion of curves in R[sup(3)]""; ""Exercises to Chapter 1""; ""Chapter 2. Symplectic and Pseudo-Euclidean Spaces""; ""2.1. Geometric structures in linear spaces""; ""2.1.1. Pseudo-Euclidean and symplectic spaces""; ""2.1.2. Symplectic transformations""; ""2.2. The Minkowski space""; ""2.2.1. The event space of the special relativity theory""; ""2.2.2. The Poincare group""; ""2.2.3. Lorentz transformations""; ""Exercises to Chapter 2""; ""Chapter 3. Geometry of Two-Dimensional Manifolds""

""3.1. Surfaces in three-dimensional space""""3.1.1. Regular surfaces""; ""3.1.2. Local coordinates""; ""3.1.3. Tangent space""; ""3.1.4. Surfaces as two-dimensional manifolds""; ""3.2. Riemannian metric on a surface""; ""3.2.1. The length of a curve on a surface""; ""3.2.2. Surface area""; ""3.3. Curvature of a surface""; ""3.3.1. On the notion of the surface curvature""; ""3.3.2. Curvature of lines on a surface""; ""3.3.3. Eigenvalues of a pair of scalar products""; ""3.3.4. Principal curvatures and the Gaussian curvature""; ""3.4. Basic equations of the theory of surfaces""

""3.4.1. Derivational equations as the ""zero curvature""condition. Gauge fields""""3.4.2. The Codazzi and sine-Gordon equations""; ""3.4.3. The Gauss theorem""; ""Exercises to Chapter 3""; ""Chapter 4. Complex Analysis in the Theory of Surfaces""; ""4.1. Complex spaces and analytic functions""; ""4.1.1. Complex vector spaces""; ""4.1.2. The Hermitian scalar product""; ""4.1.3. Unitary and linear-fractional transformations""; ""4.1.4. Holomorphic functions and the Cauchy�Riemann equations""; ""4.1.5. Complex-analytic coordinate changes""; ""4.2. Geometry of the sphere""

""4.2.1. The metric of the sphere""""4.2.2. The group of motions of a sphere""; ""4.3. Geometry of the pseudosphere""; ""4.3.1. Space-like surfaces in pseudo-Euclidean spaces""; ""4.3.2. The metric and the group of motions of thepseudosphere""; ""4.3.3. Models of hyperbolic geometry""; ""4.3.4. Hilbert's theorem on impossibility of imbedding thepseudosphere into R[sup(3)]""; ""4.4. The theory of surfaces in terms of a conformal parameter""; ""4.4.1. Existence of a conformal parameter""; ""4.4.2. The basic equations in terms of a conformal parameter""

""4.4.3. Hopf differential and its applications""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 621-624) and index.

Description based on print version record.

ISBN:

1-4704-2110-0