Geodesic
Rigidity of Hamiltonian Actions
Canadian mathematical bulletin, Tome 46 (2003) no. 2, pp. 277-290

Voir la notice de l'article provenant de la source Cambridge University Press

This paper studies the following question: Given an ${\omega }'$ -symplectic action of a Lie group on a manifold $M$ which coincides, as a smooth action, with a Hamiltonian $\omega$ -action, when is this action a Hamiltonian ${\omega }'$ -action? Using a result of Morse-Bott theory presented in Section 2, we show in Section 3 of this paper that such an action is in fact a Hamiltonian ${\omega }'$ -action, provided that $M$ is compact and that the Lie group is compact and connected. This result was first proved by Lalonde-McDuff-Polterovich in 1999 as a consequence of a more general theory that made use of hard geometric analysis. In this paper, we prove it using classical methods only.
DOI : 10.4153/CMB-2003-028-7
Mots-clés : 53D05, 37J25 
Rochon, Frédéric. Rigidity of Hamiltonian Actions. Canadian mathematical bulletin, Tome 46 (2003) no. 2, pp. 277-290. doi: 10.4153/CMB-2003-028-7
@article{10_4153_CMB_2003_028_7,
     author = {Rochon, Fr\'ed\'eric},
     title = {Rigidity of {Hamiltonian} {Actions}},
     journal = {Canadian mathematical bulletin},
     pages = {277--290},
     year = {2003},
     volume = {46},
     number = {2},
     doi = {10.4153/CMB-2003-028-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2003-028-7/}
}
TY  - JOUR
AU  - Rochon, Frédéric
TI  - Rigidity of Hamiltonian Actions
JO  - Canadian mathematical bulletin
PY  - 2003
SP  - 277
EP  - 290
VL  - 46
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2003-028-7/
DO  - 10.4153/CMB-2003-028-7
ID  - 10_4153_CMB_2003_028_7
ER  - 
%0 Journal Article
%A Rochon, Frédéric
%T Rigidity of Hamiltonian Actions
%J Canadian mathematical bulletin
%D 2003
%P 277-290
%V 46
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2003-028-7/
%R 10.4153/CMB-2003-028-7
%F 10_4153_CMB_2003_028_7

[1] [1] Bröcker, T. and Dieck, T. T., Representations of Compact Lie Groups. Graduate Texts in Math. 98, Springer-Verlag, New York, 1985. Google Scholar

[2] [2] Hartman, P., Ordinary differential equations. Wiley, New York, 1964. Google Scholar

[3] [3] Lalonde, F. and McDuff, D., Cohomological properties of ruled symplectic structures. In: Mirror symmetry and string geometry, CRM Lecture Notes and Proceedings (eds. E. D'Hoker, D. Phong and S. T. Yau), Proceedings of the workshop onMirror symmetry and string geometry (March 2000, CRM, Montreal), American Mathematical Society, 2001, to appear. Google Scholar

[4] [4] McDuff, D. and Salamon, D., Introduction to Symplectic Topology. Oxford Science Publications, 1995. Google Scholar

[5] [5] Milnor, J., Morse Theory. Ann. of Math. Stud. 51, Princeton University Press, 1963. Google Scholar

[6] [6] Quan, Pham Mau, Introduction à la géométrie des variétés différentiables. Dunod, Paris, 1969. Google Scholar

Cité par Sources :