Generalized Swan's theorem and its application
Author:
P. Manoharan
Journal:
Proc. Amer. Math. Soc. 123 (1995), 32193223
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 nonlinear differential operators of degree r under the equivalence of Swan's theorem and thus generalize Swan's theorem to include nonlinear 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]
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 (83j:58014), http://dx.doi.org/10.1090/S027309791982150042
 [2]
Shoshichi
Kobayashi, Manifolds over function algebras and mapping
spaces, Tohoku Math. J. (2) 41 (1989), no. 2,
263–282. MR
996015 (90g:58014), http://dx.doi.org/10.2748/tmj/1178227825
 [3]
P. Manoharan, A nonlinear version of Swan's theorem, Math. Z. 209 (1992), 467479.
 [4]
Richard
G. Swan, Vector bundles and projective
modules, Trans. Amer. Math. Soc. 105 (1962), 264–277. MR 0143225
(26 #785), http://dx.doi.org/10.1090/S00029947196201432256
 [1]
 R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 65222. MR 656198 (83j:58014)
 [2]
 S. Kobayashi, Manifolds over function algebras and mapping spaces, Tôkoku Math. J. (2) 41 (1989), 263282. MR 996015 (90g:58014)
 [3]
 P. Manoharan, A nonlinear version of Swan's theorem, Math. Z. 209 (1992), 467479.
 [4]
 R. G. Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264277. MR 0143225 (26:785)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919951264823X
PII:
S 00029939(1995)1264823X
Keywords:
Nonlinear differential operators,
map,
manifold
Article copyright:
© Copyright 1995
American Mathematical Society
