Manifolds of Differentiable Mappings

Manifolds of Differentiable Mappings
Title Manifolds of Differentiable Mappings PDF eBook
Author Peter W. Michor
Publisher
Pages 176
Release 1980
Genre Mathematics
ISBN

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Topology from the Differentiable Viewpoint

Topology from the Differentiable Viewpoint
Title Topology from the Differentiable Viewpoint PDF eBook
Author John Willard Milnor
Publisher Princeton University Press
Pages 80
Release 1997-12-14
Genre Mathematics
ISBN 9780691048338

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This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem.

Singularities of Differentiable Maps

Singularities of Differentiable Maps
Title Singularities of Differentiable Maps PDF eBook
Author V.I. Arnold
Publisher Springer Science & Business Media
Pages 390
Release 2012-12-06
Genre Mathematics
ISBN 1461251540

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... there is nothing so enthralling, so grandiose, nothing that stuns or captivates the human soul quite so much as a first course in a science. After the first five or six lectures one already holds the brightest hopes, already sees oneself as a seeker after truth. I too have wholeheartedly pursued science passionately, as one would a beloved woman. I was a slave, and sought no other sun in my life. Day and night I crammed myself, bending my back, ruining myself over my books; I wept when I beheld others exploiting science fot personal gain. But I was not long enthralled. The truth is every science has a beginning, but never an end - they go on for ever like periodic fractions. Zoology, for example, has discovered thirty-five thousand forms of life ... A. P. Chekhov. "On the road" In this book a start is made to the "zoology" of the singularities of differentiable maps. This theory is a young branch of analysis which currently occupies a central place in mathematics; it is the crossroads of paths leading from very abstract corners of mathematics (such as algebraic and differential geometry and topology, Lie groups and algebras, complex manifolds, commutative algebra and the like) to the most applied areas (such as differential equations and dynamical systems, optimal control, the theory of bifurcations and catastrophes, short-wave and saddle-point asymptotics and geometrical and wave optics).

Topological Library

Topological Library
Title Topological Library PDF eBook
Author Sergeĭ Petrovich Novikov
Publisher World Scientific
Pages 278
Release 2010
Genre Mathematics
ISBN 981283687X

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1. On manifolds homeomorphic to the 7-sphere / J. Milnor -- 2. Groups of homotopy spheres. I / M. Kervaire and J. Milnor -- 3. Homotopically equivalent smooth manifolds / S.P. Novikov -- 4. Rational Pontrjagin classes. Homeomorphism and homotopy type of closed manifolds / S.P. Novikov -- 5. On manifolds with free abelian fundamental group and their applications (Pontrjagin classes, smooth structures, high-dimensional knots) / S.P. Novikov -- 6. Stable homeomorphisms and the annulus conjecture / R. Kirby

Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers

Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers
Title Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers PDF eBook
Author P.M. Gadea
Publisher Springer Science & Business Media
Pages 446
Release 2009-12-12
Genre Mathematics
ISBN 9048135648

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A famous Swiss professor gave a student’s course in Basel on Riemann surfaces. After a couple of lectures, a student asked him, “Professor, you have as yet not given an exact de nition of a Riemann surface.” The professor answered, “With Riemann surfaces, the main thing is to UNDERSTAND them, not to de ne them.” The student’s objection was reasonable. From a formal viewpoint, it is of course necessary to start as soon as possible with strict de nitions, but the professor’s - swer also has a substantial background. The pure de nition of a Riemann surface— as a complex 1-dimensional complex analytic manifold—contributes little to a true understanding. It takes a long time to really be familiar with what a Riemann s- face is. This example is typical for the objects of global analysis—manifolds with str- tures. There are complex concrete de nitions but these do not automatically explain what they really are, what we can do with them, which operations they really admit, how rigid they are. Hence, there arises the natural question—how to attain a deeper understanding? One well-known way to gain an understanding is through underpinning the d- nitions, theorems and constructions with hierarchies of examples, counterexamples and exercises. Their choice, construction and logical order is for any teacher in global analysis an interesting, important and fun creating task.

An Introduction To Differential Manifolds

An Introduction To Differential Manifolds
Title An Introduction To Differential Manifolds PDF eBook
Author Dennis Barden
Publisher World Scientific
Pages 231
Release 2003-03-12
Genre Mathematics
ISBN 1911298232

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This invaluable book, based on the many years of teaching experience of both authors, introduces the reader to the basic ideas in differential topology. Among the topics covered are smooth manifolds and maps, the structure of the tangent bundle and its associates, the calculation of real cohomology groups using differential forms (de Rham theory), and applications such as the Poincaré-Hopf theorem relating the Euler number of a manifold and the index of a vector field. 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.

Weakly Differentiable Mappings between Manifolds

Weakly Differentiable Mappings between Manifolds
Title Weakly Differentiable Mappings between Manifolds PDF eBook
Author Piotr Hajłasz
Publisher American Mathematical Soc.
Pages 88
Release 2008
Genre Mathematics
ISBN 0821840797

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The authors study Sobolev classes of weakly differentiable mappings $f: {\mathbb X}\rightarrow {\mathbb Y}$ between compact Riemannian manifolds without boundary. These mappings need not be continuous. They actually possess less regularity than the mappings in ${\mathcal W}{1, n}({\mathbb X}\, \, {\mathbb Y})\, $, $n=\mbox{dim}\, {\mathbb X}$. The central themes being discussed a