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A generalisation of the Morse inequalities

Part of: Symplectic geometry, contact geometry Global differential geometry

Published online by Cambridge University Press:  09 April 2009

Mohan Bhupal
Mohan Bhupal
Affiliation:
Mathematics Department Middle East Technical University06531 AnkaraTurkey e-mail: bhupal@math.metu.edu.tr
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Abstract

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In this paper we construct a family of variational families for a Legendrian embedding, into the 1-jet bundle of a closed manifold, that can be obtained from the zero section through Legendrian embdeddings, by discretising the action functional. We compute the second variation of a generating funciton obtained as above at a nondegenerate critical point and prove a formula relating the signature of the second variation to the Maslov index as the mesh goes to zero. We use this to prove a generlisation of the Morse inequalities thus refining a theorem of Chekanov.

MSC classification

Secondary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 53D35: Global theory of symplectic and contact manifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 70 , Issue 3 , June 2001 , pp. 351 - 386
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Arnold, V. I., ‘First steps in symplectic topology’, Russian Math. Surveys 41 (1986), 1–21.CrossRefGoogle Scholar
[2]Brunella, M., ‘On a theorem of Sikorav’, Enseign. Math. (2) 37 (1991), 83–87.Google Scholar
[3]Chaperon, M., ‘Une idée du type ‘géodésiques brisées’ pour les systemes Hamiltoniens’, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), 293–296.Google Scholar
[4]Chaperon, M., ‘On generating families’, in:The Floer Memorial Volume (eds. Hofer, H., Taubes, C., Weinstein, A. and Zehnder, E.) (Birkäuser, Basel, 1996) pp. 283–296.Google Scholar
[5]Chekanov, Yu. V., ‘Critical points of quasi-functions and generating families of Legendrian manifolds’, Functional Anal. Appl. 30 (1996), 118–128.CrossRefGoogle Scholar
[6]Conley, C. C., Isolated invariant sets and the Morse index, CBMS, Reg. Conf. Ser. in Math. 38 (American Math. Soc., Providence, RI, 1978).CrossRefGoogle Scholar
[7]Conley, C. C. and Zehnder, E., ‘The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnold’, Invent. Math. 73 (1983), 33–49.CrossRefGoogle Scholar
[8]Duistermaat, J. J., ‘On the Morse index theorem in variational calculus’, Adv. Math. 21 (1976), 173–195.CrossRefGoogle Scholar
[9]Hofer, H., ‘Lagrangian embeddings and critical point theory’, Ann. Inst. H. Poincaré Anal. NonLinéaire 2 (1985), 407–462.CrossRefGoogle Scholar
[10]Hörmander, L., ‘Fourier integral operators I’, Acta Math. 127 (1971), 79–183.CrossRefGoogle Scholar
[11]Laudenbach, F. and Sikorav, J. C., ‘Persistance d'intersection avec la section nulle au cours d'une isotopie Hamiltonienne dans un fibré cotangent’, Invent. Math. 82 (1985), 349–357.CrossRefGoogle Scholar
[12]McDuff, D. and Salamon, D., Introduction to symplectic topology (Oxford University Press, Oxford, 1995).CrossRefGoogle Scholar
[13]Robbin, J. and Salamon, D., ‘The Maslov index for paths’, Topology 32 (1993), 827–844.CrossRefGoogle Scholar
[14]Robbin, J. and Salamon, D., ‘Feynman path integrals on the phase space and the metaplectic representation’, Math. Z. 221 (1996), 307–335.CrossRefGoogle Scholar
[15]Sikorav, J. C., ‘Problèmes d'intersection et de points fixes en géométrie Hamiltonienne’, Comm. Math. Helvet. 62 (1987), 62–73.CrossRefGoogle Scholar
[16]Théret, D., Utilisation des fonctions génératrices en géométrie symplectique globale (Ph.D. Thesis, Université Denis Diderot, Paris 7, 1995).Google Scholar
[17]Viterbo, C., ‘Intersection de sous-variétés Lagrangiennes, fonctionnelles d'action et indice des systèmes Hamiltoniens’, Bull. Soc. Math. France 115 (1987), 361–390.CrossRefGoogle Scholar
[18]Viterbo, C., ‘Symplectic topology as the geometry of generating functions’, Math. Ann. 292 (1992), 685–710.CrossRefGoogle Scholar