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Connections on infinite-dimensional manifolds with corners. (English) Zbl 0898.53022

This paper extends the study of infinite dimensional manifolds with corners, see, e.g., the earlier works by the authors [S. Armas-Gómez, J. Margalef Roig, E. Outerelo-Domínguez and E. Padrón-Fernández, Rend. Circ. Mat. Palermo, II. Ser. 44, 45-93 (1995; Zbl 0833.58007); the last three authors, Acta Math. Hung. 72, 105-119 (1996; Zbl 0865.58004)]. The connections on general fiber bundles and vector bundles over such manifolds are the main theme here. About one half of the paper is devoted to the exposition of basic categorical constructions with such bundles; then various types of connections are discussed. As usual in theories including infinite dimensional manifolds, the exposition must be as much coordinate free as possible and then most of the standard features from the finite dimensional cases can be recovered for Banach spaces. The existence theorems are often exceptions, though. The authors prove the existence of linear connections and principal connections at the end of their paper.

MSC:

53C05 Connections (general theory)
58B10 Differentiability questions for infinite-dimensional manifolds
57N20 Topology of infinite-dimensional manifolds
57R22 Topology of vector bundles and fiber bundles
58E40 Variational aspects of group actions in infinite-dimensional spaces
55R05 Fiber spaces in algebraic topology
55R10 Fiber bundles in algebraic topology
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References:

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