Bi-invariant metrics on the group of symplectomorphisms

Abstract

This paper studies the extension of the Hofer metric and general Finsler metrics on Hamiltonian symplectomorphism group $Ham(M,\omega)$ to the identity component $Symp_0(M,\omega)$ of symplectomorphism group. In particular, we prove that the Hofer metric on $Ham(M,\omega)$ does not extend to a bi-invariant metric on $Symp_0(M,\omega)$ for many symplectic manifolds. We also show that for the torus $T^{2n}$ with the standard symplectic form $\omega$, no Finsler metric on $Ham(T^{2n},\omega)$ that satisfies a strong form of the invariance condition can extend to a bi-invariant metric on $Symp_0(T^{2n},\omega)$. Another interesting result is that there exists no $C^1$-continuous bi-invariant metric on $Symp_0(T^{2n},\omega)$.