By Loring W. Tu

this article provides a graduate-level creation to differential geometry for arithmetic and physics scholars. The exposition follows the ancient improvement of the ideas of connection and curvature with the objective of explaining the Chern–Weil thought of attribute sessions on a primary bundle. Along the best way we stumble upon many of the excessive issues within the heritage of differential geometry, for instance, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. routines through the e-book try out the reader’s realizing of the fabric and occasionally illustrate extensions of the idea. firstly, the necessities for the reader comprise a passing familiarity with manifolds. After the 1st bankruptcy, it turns into essential to comprehend and manage differential varieties. an information of  de Rham cohomology is needed for the final 3rd of the text.

Prerequisite fabric is contained in author's text An advent to Manifolds, and will be discovered in a single semester. For the advantage of the reader and to set up universal notations, Appendix A remembers the fundamentals of manifold conception. also, in an try and make the exposition extra self-contained, sections on algebraic buildings resembling the tensor product and the outside strength are included.

Differential geometry, as its identify implies, is the learn of geometry utilizing differential calculus. It dates again to Newton and Leibniz within the 17th century, however it used to be now not until eventually the 19th century, with the paintings of Gauss on surfaces and Riemann at the curvature tensor, that differential geometry flourished and its glossy starting place used to be laid. during the last 100 years, differential geometry has confirmed necessary to an figuring out of the actual global, in Einstein's normal idea of relativity, within the concept of gravitation, in gauge thought, and now in string conception. Differential geometry can also be worthy in topology, numerous advanced variables, algebraic geometry, advanced manifolds, and dynamical structures, between different fields.  The box has even discovered functions to team idea as in Gromov's paintings and to likelihood idea as in Diaconis's paintings. it's not too far-fetched to argue that differential geometry may be in each mathematician's arsenal.

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Differential Geometry: Connections, Curvature, and Characteristic Classes (Graduate Texts in Mathematics) by Loring W. Tu

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