  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 $Symp_0(T^{2n},\omega)$.
