Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T10:47:10.724Z Has data issue: false hasContentIssue false
  • English
  • Français

Almost-Riemannian Structures on Banach Manifolds: The Morse Lemma and the Darboux Theorem

Published online by Cambridge University Press:  20 November 2018

A. J. Tromba
A. J. Tromba*
Affiliation:
University of California, Santa Cruz, California
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we introduce the notion of an almost Riemannian manifold. Briefly speaking, an almost-Riemannian structure on a Banach manifold is a generalization of the notion of a Riemannian structure on a Hilbert manifold, common examples would be manifolds of maps modelled on the Sobolev spaces Lkp. The successful use of weak Riemannian structures in hard problems has been given by Ebin [2] and Ebin and Marsden [10].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 3 , 01 June 1976 , pp. 640 - 652
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Calderon, A. P., Lebesgue spaces of differentiable functions and distributions, AMS Symposia in Pure Math, Vol. 4, (1966).Google Scholar
2. Ebin, D., The spaces of Riemannian metrics, Proc. Symp. Pure Math (1970), 11–40.Google Scholar
3. Eells, J., A setting for global analysis, Bull. AMS Sept. (1966), 751–807.Google Scholar
4. Eliasson, H. I., Variation integrals in fibre bundles, AMS Proc. Symp. in Pure Math. 16 (1970), 67–90.Google Scholar
5. Friedman, A., Partial differential equations (Holt, Rinehart, Winston, 1969).Google Scholar
6. Gromoll, D. and Meyer, W., On differential functions with isolated critical points, Topology 8 (1969), 361–369.Google Scholar
7. Gromoll, D. and Meyer, W., Periodic geodesies on compact Riemannian manifolds, J. Diff. Geom. 3 (1969), 493–510.Google Scholar
8. Lang, L., Introduction to differential manifolds (Interscience, 1962).Google Scholar
9. Marsden, J., Darbouxs theorem fails for weak simplectic forms, Proc. AMS 32 (1972), 590–592.Google Scholar
10. Marsden, J. and Ebin, D., Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970), 102–163.Google Scholar
11. Palais, R., Foundations of global non-linear analysis (Benjamin, 1968).Google Scholar
12. Palais, R., Morse theory on Hilbert manifolds, Topology 2 (1963), 240–299.Google Scholar
13. Palais, R., Morse lemma on Banach spaces, Bull. Amer. Math. Soc. 75 (1969), 968–971.Google Scholar
14. Palais, R., Lusternik-Schnirelman category theory on Banach manifolds, Topology 5 (1966), 115–132.Google Scholar
15. Tromba, A., The Morse lemma on Banach spaces, Proc. Amer. Math. Soc. 34 (1972), 396–402.Google Scholar
16. Tromba, A., The Morse lemma on arbitrary Banach spaces, Bull. AMS 79 (1973), 85–86.Google Scholar
17. Tromba, A., 77jg Ruler-characteristic of vector fields on Banach manifolds and a globalization of Leray-Schauder degree, to appear, Advances in Mathematics.Google Scholar
18. Tromba, A., A general approach to Morse theory, to appear, J. Diff. Geometry.Google Scholar
19. Tromba, A. and Elworthy, K. D., Differentiable structures and Fredholm maps on Banach manifolds, Proc. Symp. on Pure Math. 15 (1970), 45–96.Google Scholar
20. Weinstein, A., Symplectic structures on Banach manifolds, Bull. Amer. Math. Soc. 75 (1969), 1040–1041.Google Scholar