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Topological algebras and abstract differential geometry. (English) Zbl 0936.53022

The notions of connection and curvature on principal sheaves, with structural sheaf the sheaf of groups \({\mathcal G}{\mathcal L}(n, {\mathcal A})\), are studied where \({\mathcal A}\) is a sheaf of unital, commutative and associative algebras. Suitable topological algebras provide concrete models of principal sheaves for which an abstract Frobenius integrability condition holds, thus establishing the equivalence between flatness, parallelism and integrability of a connection on them. Some forthcoming papers of the author on this theory are announced.

MSC:

53C05 Connections (general theory)
55R65 Generalizations of fiber spaces and bundles in algebraic topology
58A40 Differential spaces
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
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