A metric approach to Fréchet geometry

Abstract

a b s t r a c tThe aim of this article is to present the category of bounded Fréchet manifolds in respect to which we will review the geometry of Fréchet manifolds with a stronger accent on its metric aspect. An inverse function theorem in the sense of Nash and Moser in this category is proved, and some examples from Riemannian geometry are given.

Metrizability of spaces of homomorphisms between metric vector spaces

“…[6] where it is shown that this is a bounded-smooth map). Obviously, for r < s and k ≤ ∞, we have always inclusions…”

“…Remark: Property 5 is needed only to show Hausdorffness. Thus, for example, B r L(V, W ) as described in [6] does not come from any palette, even without 5, as it is Hausdorff anyway.…”

Sard’s theorem for mappings between Fréchet manifolds

“…If these Fréchet spaces are endowed with Fréchet metrics, then we require these metrics to be standard and the transition functions to be globally Lipschitz and M C 1 : In this case, the manifold is called a bounded Fréchet manifold. If a Fréchet manifold carries a compatible metric, then it is bounded ( [4], Theorem 3.33). Moreover, since we need manifolds to carry compatible metrics, we only deal with bounded Fréchet manifolds.…”

“…In addition, it is necessary to restrict the class of maps under consideration to the maps for which the inverse-function theorem is available (this theorem is required to prove a local representation theorem for Lipschitz-Fredholm maps, which plays an important role in proving the main theorem). The analyzed class of maps is the class of so-called M C k -maps introduced in [4,5]. Indeed, as already indicated, there exists a suitable topology in the space of linear globally Lipschitz continuous maps and an inverse-function theorem was obtained in this category (see [4,5]).…”

“…The analyzed class of maps is the class of so-called M C k -maps introduced in [4,5]. Indeed, as already indicated, there exists a suitable topology in the space of linear globally Lipschitz continuous maps and an inverse-function theorem was obtained in this category (see [4,5]). …”