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

MathSciNet review:
1264823

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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 *k*th derivatives are linear differential operators of degree *r* in each variable, is given in terms of -maps.

**[1]**Richard S. Hamilton,*The inverse function theorem of Nash and Moser*, Bull. Amer. Math. Soc. (N.S.)**7**(1982), no. 1, 65–222. MR**656198**, 10.1090/S0273-0979-1982-15004-2**[2]**Shoshichi Kobayashi,*Manifolds over function algebras and mapping spaces*, Tohoku Math. J. (2)**41**(1989), no. 2, 263–282. MR**996015**, 10.2748/tmj/1178227825**[3]**P. Manoharan,*A non-linear version of Swan's theorem*, Math. Z.**209**(1992), 467-479.**[4]**Richard G. Swan,*Vector bundles and projective modules*, Trans. Amer. Math. Soc.**105**(1962), 264–277. MR**0143225**, 10.1090/S0002-9947-1962-0143225-6

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1995-1264823-X

Keywords:
Non-linear differential operators,
-map,
-manifold

Article copyright:
© Copyright 1995
American Mathematical Society