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The Metric Theory of Banach Manifolds (Lecture Notes in Mathematics)
The Metric Theory of Banach Manifolds (Lecture Notes in Mathematics)
Date: 21 June 2011, 01:28

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Compactness is frequently an annoying hypothesis in differential topology. Even when one is primarily interested in a compact manifold, associated noncompact manifolds turn up, eg. the leaf space of a foliation. Also, in technical constructions, it would be helpful to be able to dispense with compactness. For example, if f is a diffeomorphism of a compact manifold, X, then it is helpful in studying the dynamics of f to regard the integers, Z, as a discrete manifold and look at the manifold of maps, ?(Z,X). Again it is compactness that prevents one from noting that Hartman's Theorem is not only related to the structural stability theorem for Anosov diffeomorphisms but is in fact a corollary of the latter because a hyperbolic linear map is an Anosov diffeomorphism of Euclidean space.

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