000			03260nam a2200325Ia 4500
003			OSt
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020			_a9783319855622 (pbk.) _c€ 54.99
040			_bENG _cIISER-BPR
041			_aENG
082			_a516.36 _bLOR _223rd
100			_aLoring W. Tu _91070
222			_aMathematics
245		0	_aDifferential geometry : _bConnections, curvature, and characteristic classes
250			_a1st ed.
260			_aCham, Switzerland: _bSpringer Nature, _cc2017.
300			_axzvii, 346p. : _bill(col.). ; _c24cm
440			_aGraduate Texts in Mathematics _vVol. 275 _92545
504			_aIncludes appendices, list of notations, Bibliographic references and subject index.
520			_aThis 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.
650			_aMathematics _92546
650			_aGeometry _92547
650			_aAnalytic geometries _92548
650			_aDifferential geometry _92549
942			_cBK _2ddc
942			_2ddc
947			_a5039.833500000001
948			_a22
999			_c4210 _d4210