At the most basic level, the book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. In differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differentiable maps in them immersions, embeddings, isomorphisms, etc. Home Differential and Riemannian Manifolds. Genre : Mathematics Get Book. Riemannian Manifolds Author : John M.
Conner Publisher : American Mathematical Soc. Genre : Differential forms Get Book. Popular Books. Each chapter contains exercises of varying difficulty for which solutions are provided. Special features include examples drawn from geometric manifolds in dimension 3 and Brieskorn varieties in dimensions 5 and 7, as well as detailed calculations for the cohomology groups of spheres and tori. I have expanded the book considerably, including things like the Lie derivative, and especially the basic integration theory of differential forms, with Stokes' theorem and its various special formulations in different contexts.
The foreword which I wrote in the earlier book is still quite valid and needs only slight extension here.
Between advanced calculus and the three great differential theories differential topology, differential geometry, ordinary differential equations , there lies a no-man's-land for which there exists no systematic exposition in the literature. In differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differentiable maps in them immersions, embeddings, isomorphisms, etc.
One may also use differentiable structures on topological manifolds to determine the topological structure of the manifold e. Complete, detailed treatment, enhanced with philosophical and historical asides and more than exercises.
This book is an introduction to differential manifolds. It gives solid preliminaries for more advanced topics: Riemannian manifolds, differential topology, Lie theory.
It presupposes little background: the reader is only expected to master basic differential calculus, and a little point-set topology. The book covers the main topics of differential geometry: manifolds, tangent space, vector fields, differential forms, Lie groups, and a few more sophisticated topics such as de Rham cohomology, degree theory and the Gauss-Bonnet theorem for surfaces. Its ambition is to give solid foundations.
More than exercises, some of them easy and classical, some others more sophisticated, will help the beginner as well as the more expert reader. Solutions are provided for most of them. The book should be of interest to various readers: undergraduate and graduate students for a first contact to differential manifolds, mathematicians from other fields and physicists who wish to acquire some feeling about this beautiful theory.
Jacques Lafontaine was successively assistant Professor at Paris Diderot University and Professor at the University of Montpellier, where he is presently emeritus. His main research interests are Riemannian and pseudo-Riemannian geometry, including some aspects of mathematical relativity. Besides his personal research articles, he was involved in several textbooks and research monographs. It is addressed primarily to second year graduate students and well prepared first year students.
Presupposed is a good grounding in general topology and modern algebra, especially linear algebra and the analogous theory of modules over a commutative, unitary ring.
Although billed as a "first course" , the book is not intended to be an overly sketchy introduction. Mastery of this material should prepare the student for advanced topics courses and seminars in differen tial topology and geometry. There are certain basic themes of which the reader should be aware.
The first concerns the role of differentiation as a process of linear approximation of non linear problems.
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