Characterization of diffeomorphisms that are symplectomorphisms

Stanisław Janeczko¹ ; Zbigniew Jelonek²

¹ S. Janeczko Instytut Matematyczny PAN Śniadeckich 8 00-956 Warszawa, Poland and Wydział Matematyki i Nauk Informacyjnych Politechnika Warszawska Pl. Politechniki 1 00-661 Warszawa, Poland
² Z. Jelonek Instytut Matematyczny PAN Śniadeckich 8 00-956 Warszawa, Poland

Fundamenta Mathematicae, Tome 205 (2009) no. 2, pp. 147-160.

Résumé

Let $(X , \omega _X)$ and $(Y, \omega _Y)$ be compact symplectic manifolds (resp. symplectic manifolds) of dimension $2n>2.$ Fix $ 0 s n$ (resp. $ 0 k\leq n$) and assume that a diffeomorphism ${\Phi } : X \to Y$ maps all $2s$-dimensional symplectic submanifolds of $X$ to symplectic submanifolds of $Y$ (resp. all isotropic $k$-dimensional tori of $X$ to isotropic tori of $Y$). We prove that in both cases ${\Phi }$ is a conformal symplectomorphism, i.e., there is a constant $c\not =0$ such that ${ \Phi }^*\omega _Y=c\omega _X.$

Détail

DOI : 10.4064/fm205-2-4

Keywords: omega omega compact symplectic manifolds resp symplectic manifolds dimension fix resp leq assume diffeomorphism phi maps s dimensional symplectic submanifolds symplectic submanifolds resp isotropic k dimensional tori isotropic tori prove cases phi conformal symplectomorphism there constant phi * omega omega