Generalized Swan's theorem and its application
Author:
P. Manoharan
Journal:
Proc. Amer. Math. Soc. 123 (1995), 3219-3223
MSC:
Primary 58D15; Secondary 13C10, 55R10
DOI:
https://doi.org/10.1090/S0002-9939-1995-1264823-X
MathSciNet review:
1264823
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Swan's theorem verifies the equivalence between finitely generated projective modules over function algebras and smooth vector bundles. We define -maps that correspond to usual non-linear differential operators of degree r under the equivalence of Swan's theorem and thus generalize Swan's theorem to include non-linear differential operators as morphisms. An
-manifold structure is introduced on the space of sections of a fiber bundle through charts with
-maps as transition homeomorphisms. A characterization for all the smooth maps between the spaces of sections of vector bundles, whose kth derivatives are linear differential operators of degree r in each variable, is given in terms of
-maps.
- [1] R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 65-222. MR 656198 (83j:58014)
- [2] S. Kobayashi, Manifolds over function algebras and mapping spaces, Tôkoku Math. J. (2) 41 (1989), 263-282. MR 996015 (90g:58014)
- [3] P. Manoharan, A non-linear version of Swan's theorem, Math. Z. 209 (1992), 467-479.
- [4] R. G. Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264-277. MR 0143225 (26:785)
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58D15, 13C10, 55R10
Retrieve articles in all journals with MSC: 58D15, 13C10, 55R10
Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1995-1264823-X
Keywords:
Non-linear differential operators,
-map,
-manifold
Article copyright:
© Copyright 1995
American Mathematical Society