Connections, Curvature, and Cohomology: Lie groups, principal bundles, and characteristic classes

Connections, Curvature, and Cohomology: Lie groups, principal bundles, and characteristic classes
Title Connections, Curvature, and Cohomology: Lie groups, principal bundles, and characteristic classes PDF eBook
Author Werner Hildbert Greub
Publisher
Pages 572
Release 1973
Genre Mathematics
ISBN

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Volume 2.

Connections, Curvature and Cohomology: De Rham cohomology of manifolds and vector bundles ; Vol. 2, Lie groups, principal bundles and characteristic classes

Connections, Curvature and Cohomology: De Rham cohomology of manifolds and vector bundles ; Vol. 2, Lie groups, principal bundles and characteristic classes
Title Connections, Curvature and Cohomology: De Rham cohomology of manifolds and vector bundles ; Vol. 2, Lie groups, principal bundles and characteristic classes PDF eBook
Author Werner Greub
Publisher
Pages 984
Release 1972
Genre
ISBN 9780123027016

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Spectral Theory of Random Matrices

Spectral Theory of Random Matrices
Title Spectral Theory of Random Matrices PDF eBook
Author Vyacheslav L. Girko
Publisher Academic Press
Pages 568
Release 2016-08-23
Genre Computers
ISBN 0080873618

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Spectral Theory of Random Matrices

Differential Geometry

Differential Geometry
Title Differential Geometry PDF eBook
Author Loring W. Tu
Publisher Springer
Pages 358
Release 2017-06-01
Genre Mathematics
ISBN 3319550845

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This text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included. Differential geometry, as its name implies, is the study of geometry using differential calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal.

Connections, Curvature, and Cohomology V1

Connections, Curvature, and Cohomology V1
Title Connections, Curvature, and Cohomology V1 PDF eBook
Author
Publisher Academic Press
Pages 467
Release 1972-07-31
Genre Mathematics
ISBN 008087360X

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Connections, Curvature, and Cohomology V1

Characteristic Classes

Characteristic Classes
Title Characteristic Classes PDF eBook
Author John Willard Milnor
Publisher Princeton University Press
Pages 342
Release 1974
Genre Mathematics
ISBN 9780691081229

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The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds. In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class. Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli numbers. Based on lecture notes of John Milnor, which first appeared at Princeton University in 1957 and have been widely studied by graduate students of topology ever since, this published version has been completely revised and corrected.

Connections, Curvature, and Cohomology

Connections, Curvature, and Cohomology
Title Connections, Curvature, and Cohomology PDF eBook
Author Werner Hildbert Greub
Publisher Academic Press
Pages 618
Release 1972
Genre Mathematics
ISBN 0123027039

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This monograph developed out of the Abendseminar of 1958-1959 at the University of Zürich. The purpose of this monograph is to develop the de Rham cohomology theory, and to apply it to obtain topological invariants of smooth manifolds and fibre bundles. It also addresses the purely algebraic theory of the operation of a Lie algebra in a graded differential algebra.