Characterization of diffeomorphisms that are symplectomorphisms
Tom 205 / 2009
Fundamenta Mathematicae 205 (2009), 147-160
MSC: 53D05, 51N10, 53D22, 70H05, 15A04.
DOI: 10.4064/fm205-2-4
Streszczenie
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 ${ \Phi }^*\omega _Y=c\omega _X.$
