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Differential forms with values in groups

Published online by Cambridge University Press:  17 April 2009

Anders Kock
Anders Kock
Affiliation:
Matematisk Institut, Aarhus Universitet, Aarhus, Denmark.
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Abstract

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In the context of synthetic differential geometry, we present a notion of differential form with values in a group object, typically a Lie group or the group of all diffeomorphisms of a manifold. Natural geometric examples of such forms and the role of their exterior differentiation is given. The main result is a comparison with the classical theory of Lie algebra valued forms.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 25 , Issue 3 , June 1982 , pp. 357 - 386
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Dubuc, Eduardo J., “Sur les modèles de la géométrie differentielle synthétique” Cahiers Topologie Géom. Différentielle 20 (1979), 231–279.Google Scholar
[2]Ehresmann, Charles, “Les connexions infinitésimales dans un espace fibré différentiable”, Colloque de topologie (espaces fibrés), Bruxelles, 1950, 29–55 (Georges Thone, Liège; Masson, Paris; 1951).Google Scholar
[3]Ehresmann, Charles, “Catégories inductives et pseudogroupes”, Ann. Inst. Fourier (Grenoble) 10 (1960), 307–332.CrossRefGoogle Scholar
[4]Griffiths, P., “On Cartans' method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry1”, Duke Math. J. 41 (1974), 775–814.Google Scholar
[5]Haefliger, André, “Cohomology of Lie algebras and foliations”, Differential topology, foliations and Gelfand-Fuks cohomology, 1–12 (Proc. Sympos. Pontifica Universidade Católica do Rio de Janeiro, 1976. Lecture Notes in Mathematics, 652. Springer-Verlag, Berlin, Heidelberg, New York, 1978).CrossRefGoogle Scholar
[6]Hermann, Robert, “On the differential geometry of foliations”, Ann. of Math. 72 (1960), 445–457.Google Scholar
[7]Kock, Anders, “Taylor series calculus for ring objects of line type”, J. Pure Appl. Algebra 12 (1978), 271–293.Google Scholar
[8]Kock, Anders, “On the synthetic theory of vector fields”, Topos theoretic methods in geometry, 139–157 (Various Publications Series, 30. Matematisk Institut, Aarhus Universitet, Aarhus, 1979).Google Scholar
[9]Kock, Anders, “Properties of well-adapted models for synthetic differential geometry”, J. Pure Appl. Algebra 20 (1981), 55–70.CrossRefGoogle Scholar
[10]Kock, Anders, “Formal manifolds and synthetic theory of jet bundles”, Cahiers Topologie Géom. Différentielle 21 (1980), 227–246.Google Scholar
[11]Kock, Anders, Synthetic differential geometry (London Mathematical Society Lecture Notes, 51. Cambridge University Press, Cambridge, 1981).Google Scholar
[12]Kock, A. and Reyes, G.E., “Models for synthetic integration theory”, Math. Scand. 48 (1981), 145–152.Google Scholar
[13]Kock, A., Reyes, G.E., Veit, and B., Forms and integration in synthetic differential geometry (Aarhus Preprint Series, 31. Matematisk Institut, Aarhus Universitet, Aarhus, 1979/1980).Google Scholar
[14]Reyes, G.E. and Wraith, G.C., A note on tangent bundles in a category with a ring object”, Math. Scand. 42 (1978), 53–63.Google Scholar