Modern geometric structures and fields
著者
書誌事項
Modern geometric structures and fields
（Graduate studies in mathematics, v. 71）
American Mathematical Society, c2006
- タイトル別名
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Современные геометрические структуры и поля
Sovremennye geometricheskie struktury i poli︠a︡
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注記
Includes bibliographical references (p. 621-624) and index
内容説明・目次
内容説明
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.
目次
Cartesian spaces and Euclidean geometry Symplectic and pseudo-Euclidean spaces Geometry of two-dimensional manifolds Complex analysis in the theory of surfaces Smooth manifolds Groups of motions Tensor algebra Tensor fields in analysis Analysis of differential forms Connections and curvature Conformal and complex geometries Morse theory and Hamiltonian formalism Poisson and Lagrange manifolds Multidimensional variational problems Geometric fields in physics Bibliography Index.
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