Manifolds Of Differentiable Mappings

Author :
Publisher : American Mathematical Soc.
ISBN 13 : 9780821866405
Total Pages : 92 pages
Book Rating : 4.8/5 (664 download)

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Download or read book Weakly Differentiable Mappings Between Manifolds written by and published by American Mathematical Soc.. This book was released on 2008-02-15 with total page 92 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 are: smooth approximation of those mappings integrability of the Jacobian determinant The approximation problem in the category of Sobolev spaces between manifolds ${\mathcal W}^{1,p}({\mathbb X}\, ,\, {\mathbb Y})$, $1\leqslant p \leqslant n$, has been recently settled. However, the point of the results is that the authors make no topological restrictions on manifolds ${\mathbb X}$ and ${\mathbb Y}$. They characterize, essentially all, classes of weakly differentiable mappings which satisfy the approximation property. The novelty of their approach is that they were able to detect tiny sets on which the mappings are continuous. These sets give rise to the so-called web-like structure of ${\mathbb X}$ associated with the given mapping $f: {\mathbb X}\rightarrow {\mathbb Y}$. The integrability theory of Jacobians in a manifold setting is really different than one might a priori expect based on the results in the Euclidean space. To the authors' surprise, the case when the target manifold ${\mathbb Y}$ admits only trivial cohomology groups $H^\ell ({\mathbb Y})$, $1\leqslant \ell