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A metric approach to Fréchet geometry. (English) Zbl 1155.58002

The author studies the category of bounded Fréchet manifolds. He uses, instead of tame maps, some maps that ar bounded with respect to a suitable metric used on the spaces of sections. After reviewing some basic results for connections in fiber bundles, he presents the linear connections of vector bundles as Levi–Civita connections on the total space. After presenting some technical definitions and results about bounded geometry, he presents the category of bounded Fréchet manifolds. These manifolds are applied to the spaces od sections of fiber bundles. The author proves a an inverse function theorem fitting to the considered category along the lines of the proof of the inverse function theorem for maps between Banach spaces. Some examples from Riemannian geometry are given.

MSC:

58B15 Fredholm structures on infinite-dimensional manifolds
57R55 Differentiable structures in differential topology
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