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.


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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[1] D. Bleecker,Gauge Theory and Variational Principles, Addison-Wesley, Reading, Massachusetts (1981). · Zbl 0481.58002
[2] N. Bourbaki,Groupes et Algèbres de Lie, Chapitres 2–3, Hermann, Paris (1972). · Zbl 0244.22007
[3] A. Grothendieck,A General Theory of Fibre Spaces with Structural Sheaf, Kansas Univ. (1958).
[4] R. S. Gunning,Lectures on Vector Bundles over Riemann Surfaces, Princeton Univ. Press, Princeton New Jersey (1967). · Zbl 0163.31903
[5] S. Kobayashi and K. Nomizu,Foundations of Differential Geometry, Vol. 1, Interscience, New York (1963). · Zbl 0119.37502
[6] A. Mallios,Topological Algebras. Selected Topics, North-Holland, Amsterdam (1986). · Zbl 0597.46046
[7] A. Mallios, ”On an abstract form of Weil’s integrality theorem,”Note di Matematica,12, 167–202 (1992). · Zbl 0818.53028
[8] A. Mallios, ”The de Rham-Kähler complex of the Gel’fand sheaf of a topological algebra,”J. Math. Anal. Appl.,175, 143–168 (1993). · Zbl 0805.46046 · doi:10.1006/jmaa.1993.1159
[9] A. Mallios,Geometry of Vector Sheaves, Vols. I–II, Kluwer, in press.
[10] A. Mallios, ”On an axiomatic approach to geometric prequantization: a classification scheme à la Kostant-Souriau-Kirillov,” this issue. · Zbl 0939.53043
[11] M. A. Mostow, ”The differentiable structure of Milnor classifying spaces, simplicial complexes, and geometric relations,”J. Diff. Geom.,14, 255–293 (1979). · Zbl 0427.58005
[12] M. H. Papatriantafillou, ”The de Rham complex in function spaces,” In:Proc. 4th International Congress of Geometry, Thessaloniki, 1996in press. · Zbl 0885.58002
[13] I. Satake, ”On a generalization of the notion of manifold,” In:Proc. Natl. Acad. Sci. USA, Vol. 42 (1956), pp. 359–363. · Zbl 0074.18103 · doi:10.1073/pnas.42.6.359
[14] R. Sikorski, ”Differential modules,”Colloq. Mathem.,24, 45–79 (1971). · Zbl 0226.53004
[15] J. W. Smith, ”The de Rham theorem for general spaces,”Tôhoku Math. J.,18, 115–137 (1966). · Zbl 0146.19402 · doi:10.2748/tmj/1178243443
[16] E. Vassiliou, ”Connections on principal sheaves,” In:Proc. Conference on Differential Geometry, Budapest, 1996 (J. Szenthe, ed.)in press. · Zbl 0883.53029
[17] E. Vassiliou, ”On Mallios’A-connections as connections on principal sheaves,”Note di Matematicain press. · Zbl 0940.53020
[18] E. Vassiliou, ”Transformations of sheaf connections,”Balkan J. Geom. Appl.,1, 117–133 (1996). · Zbl 0877.53020
[19] E. Vassiliou, ”Flat principal sheaves,” to appear. · Zbl 0906.53018
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