An Introduction to Differential Manifolds

This publication is an creation to differential manifolds. It supplies strong preliminaries for extra complex subject matters: Riemannian manifolds, differential topology, Lie idea. It presupposes little history: the reader is just anticipated to grasp uncomplicated differential calculus, and a bit point-set topology. The e-book covers the most subject matters of differential geometry: manifolds, tangent area, vector fields, differential kinds, Lie teams, and some extra subtle subject matters similar to de Rham cohomology, measure concept and the Gauss-Bonnet theorem for surfaces.

Its ambition is to offer stable foundations. particularly, the advent of “abstract” notions reminiscent of manifolds or differential types is stimulated through questions and examples from arithmetic or theoretical physics. greater than one hundred fifty routines, a few of them effortless and classical, a few others extra refined, might help the newbie in addition to the extra professional reader. strategies are supplied for many of them.

The booklet could be of curiosity to numerous readers: undergraduate and graduate scholars for a primary touch to differential manifolds, mathematicians from different fields and physicists who desire to collect a few feeling approximately this pretty theory.

The unique French textual content advent aux variétés différentielles has been a best-seller in its classification in France for lots of years.

Jacques Lafontaine was once successively assistant Professor at Paris Diderot collage and Professor on the college of Montpellier, the place he's shortly emeritus. His major study pursuits are Riemannian and pseudo-Riemannian geometry, together with a few facets of mathematical relativity. in addition to his own learn articles, he was once serious about numerous textbooks and learn monographs.

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Universitext (Springer, 2000). J. Dieudonn ▸ Treatise on research III. natural Appl. Math. (Amst. ), vol. 10 (Academic Press, 1972). Translated from the French. Austere, yet includes many technical effects that i haven't obvious somewhere else. there are many instructive routines of average hassle (AN). ▸ A historical past of Algebraic and Differential Topology 1900–1960 (Birkaser, 1988). an exceptional connection with aid comprehend the motivations which force the notions of algebraic and differential topology; very useful position to discover the connection with “well recognized” effects (AN). M. Do Carmo ▸ Differential Geometry of Curves and Surfaces (Prentice corridor Inc. , 1976). ▸ Riemannian Geometry (Birkhaser, 1992). A. Douady and R. Douady, Algbre et thories galoisiennes, 2d ed. (Cassini, 2005). extraordinary for a truly conceptual exposition of overlaying conception (AN). B. Doubrovine, S. Novikov and A. Fomenko ▸ glossy Geometry. tools and purposes II. Grad. Texts in Math. , vol. 104 (Springer, 1985). Translated from the Russian. ▸ sleek Geometry. tools and functions III. Grad. Texts in Math. , vol. 124 (Springer, 1990). Translated from the Russian. ▸ glossy Geometry. tools and functions I, 2d ed. Grad. Texts in Math. , vol. 104 (Springer, 1992). Translated from the Russian. J. Dugundji, Topology (Allyn and Bacon, Inc. , 1965). a whole exposition of element set topology, by means of the fundamentals of homotopy concept. a bit outdated, however it is difficult to do larger! (AN). J. J. Duistermaat and J. A. Kolk, Lie teams. Universitext (Springer, 1999). H. M. Farkas and that i. Kra, Riemann Surfaces, second ed. Grad. Texts in Math. , vol. seventy one (Springer, 1991). R. Feynman, R. Leighton and M. Sands, The Feynman Lectures on Physics, the recent millenium ed. (Basic books, 1963). W. Fulton, Algebraic Topology; a primary direction. Grad. Texts in Math. , vol. 153 (Springer, 1995). a bit tougher than Greenberg-Harper eighty one, yet even more centred towards manifolds (AN). S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry (Springer, 2005). R. Godement, advent l. a. thorie des groupes de Lie (Springer, 2005). M. Golubitsky and V. Guillemin, sturdy Mappings and Their Singularities. Grad. Texts in Math. , vol. 14 (Springer, 1973). A. grey, Tubes. Progr. Math. , vol. 221 (Birkhaser, 2004). M. J. Greenberg and J. R. Harper, Lectures on Algebraic Topology. a primary direction (Benjamin/Cumming, 1981). Singular homology and cohomology, multiplicative constitution, duality. even more of a textbook than a reference e-book like [Bredon ninety four] (AN). W. Greub, S. Halperin and R. Van Stone, Connections, Curvature and Cohomology (Academic Press, 1976). to determine a scientific implementation of differential kinds (AN). P. Griffiths and J. Harris, ideas of Algebraic Geometry (John Wiley & Sons, Inc. , 1994). V. Guillemin and A. Pollack, Differential Topology (Prentice corridor Inc. , 1974). B. corridor, Lie teams, Lie Algebras and Representations. Grad. Texts in Math. , vol. 222 (Springer, 2003). R. -S. Hamilton, “The Inverse functionality Theorem of Nash and Moser”, Bull. Amer. Math. Soc. 7, pp. 65–222 (1982). a robust dialogue of the stipulations of the Banach inverse functionality theorem, and a stimulated creation to the “Nash-Moser” equipment (AN).

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