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Differential Geometry: Connections, Curvature, And Characteristic
Classes: 275
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 textPrerequisite 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 includedDifferential 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
cohomology is required for the last third of the textPrerequisite 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 includedDifferential 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
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