Manifolds of mappings and shapes
Differential Geometry
2016-04-08 v2 Functional Analysis
Abstract
This is an overview article. In his Habilitationsvortrag, Riemann described infinite dimensional manifolds parameterizing functions and shapes of solids. This is taken as an excuse to describe convenient calculus in infinite dimensions which allows for short and transparent proofs of the main facts of the theory of manifolds of smooth mappings. Smooth manifolds of immersions, diffeomorphisms, and shapes, and weak Riemannian metrics on them are treated, culminating in the surprising fact, that geodesic distance can vanish completely for them.
Keywords
Cite
@article{arxiv.1505.02359,
title = {Manifolds of mappings and shapes},
author = {Peter W. Michor},
journal= {arXiv preprint arXiv:1505.02359},
year = {2016}
}
Comments
27 pages; small changes. IN: The legacy of Bernhard Riemann after one hundred and fifty years, pp. 469--486. ALM 35, Higher Education Press and International Press Beijing--Boston. 2015