Fibrations on Banach manifolds

Abstract. For a large class of metric spaces with nice local structure, which includes Banach-Finsler manifolds and geodesic spaces of curvature bounded above, we give sufficient conditions for a local homeomorphism to be a covering projection. We first obtain a general condition in terms of a path continuation property. As a consequence, we deduce several conditions in terms of path-liftings involving a generalized derivative, and in particular we obtain an extension of Hadamard global inversion theorem in this context. Next we prove that, in the case of quasi-isometric mappings, some of these sufficient conditions are also necessary. Finally, we give some applications to the existence of global implicit functions.

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“…This extends [8,Theorem 3.5]. We note that Condition 3 was the key of the original argument used by Hadamard in [9].…”

Section: Locally Quasi-isometric Mapssupporting

confidence: 65%

“…As a direct application, using Example 3.3, we obtain the following result (compare with [8,Corollary 3.4]). …”

mentioning

confidence: 71%

Global homeomorphisms and covering projections on metric spaces

Gutú

Jaramillo

2007

Math. Ann.

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“…He introduced the concept of strong submersion with uniformly split kernels for mappings between Banach-Finsler manifolds, in terms of the corresponding analogue of µ(x). As Rabier pointed out in his previous work [52] the strong submersion condition "interpolates" two well-known, but until that moment, unrelated hypotheses corresponding to the two extreme cases: Hadamard's criterion when f : X → Y is a local diffeomorphism and the Palais-Smale condition when Y = R. Similar results can be found in [21].…”

supporting

confidence: 64%

“…Assume Y is connected. On the one hand, a simple adjustment in the proof of Earle-Eells Theorem shows that if f has the continuation property for the set of all C 1 paths then f is a fibre bundle; see Theorem 2.3 of [21]. For example, a weakly proper map has the continuation property for C 1 paths.…”

Section: −1mentioning

confidence: 99%

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On global inverse theorems

Gutú¹

2016

TMNA

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Abstract. Since the Hadamard Theorem, several metric and topological conditions have emerged in the literature to date, yielding global inverse theorems for functions in different settings. Relevant examples are the mappings between infinite-dimensional Banach-Finsler manifolds, which are the focus of this work. Emphasis is given to the nonlinear Fredholm operators of nonnegative index between Banach spaces. The results are based on good local behavior of f at every x, namely, f is a local homeomorphism or f is locally equivalent to a projection. The general structure includes a condition that ensures a global property for the fibres of f , ideally expecting to conclude that f is a global diffeomorphism or equivalent to a global projection. A review of these results and some relationships between different criteria are shown. Also, a global version of Graves Theorem is obtained for a suitable submersion f with image in a Banach space: given r > 0 and x 0 in the domain of f we give a radius ̺(r) > 0, closely related to the hypothesis of the Hadamard Theorem, such that B̺(f (x 0 )) ⊂ f (Br (x 0 )).

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A necessary condition for path-finding by the homotopy continuation method

Amiss

Guay

2009

Proceedings of the 48h IEEE Conference on Decision and Control (CDC) Held Jointly With 2009 28th Chinese Control Conference

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This paper is concerned with a path-finding method proposed by H. J. Sussmann, known as the homotopy continuation method (HCM). Given a control system initialized at some state, the HCM can be used to find an admissible open-loop control that performs a desired state transfer. This is accomplished by lifting curves in the state space of the system to curves in the space of regular admissible controls. The lifted curves are constructed as solutions of an ordinary differential equation called the path-lifting equation (PLE). The obstruction to applying the HCM, in general, is that solutions of the PLE may not be globally defined. We show that if this obstruction does not exist, then the endpoint map of the control system, restricted to the space of regular admissible controls, is necessarily a locally trivial fiber bundle. This result is valid for wide classes of control systems and controls.

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Fibrations on Banach manifolds

Cited by 5 publications

References 12 publications

Global homeomorphisms and covering projections on metric spaces

Global homeomorphisms and covering projections on metric spaces

On global inverse theorems

A necessary condition for path-finding by the homotopy continuation method

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