Bi-invariant metrics on the group of symplectomorphisms

2005/10/26 by Han, Zhigang
#53D99 #57R17 #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.math/0510569

Abstract

This paper studies the extension of the Hofer metric and general Finsler metrics on Hamiltonian symplectomorphism group Ham(M,ω) to the identity component Symp₀(M,ω) of symplectomorphism group. In particular, we prove that the Hofer metric on Ham(M,ω) does not extend to a bi-invariant metric on Symp₀(M,ω) for many symplectic manifolds. We also show that for the torus T²ⁿ with the standard symplectic form ω, no Finsler metric on Ham(T²ⁿ,ω) that satisfies a strong form of the invariance condition can extend to a bi-invariant metric on Symp₀(T²ⁿ,ω). Another interesting result is that there exists no C¹-continuous bi-invariant metric on Symp₀(T²ⁿ,ω).

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