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Modern Geometric Structures and Fields
S. P. Novikov, University of Maryland, College Park, MD, and I. A. Taimanov, Russian Academy of Sciences, Novosibirsk, Russia
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Graduate Studies in Mathematics
2006; 633 pp; hardcover
Volume: 71
ISBN-10: 0-8218-3929-2
ISBN-13: 978-0-8218-3929-4
List Price: US$81
Member Price: US$64.80
Order Code: GSM/71
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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.

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

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