This paper studies the extension of the Hofer metric and general Finsler metrics on the Hamiltonian symplectomorphism group
Ham
(
M
,
ω
)
\textrm {Ham}(M,\omega )
to the identity component
Symp
0
(
M
,
ω
)
\textrm {Symp}_0(M,\omega )
of the symplectomorphism group. In particular, we prove that the Hofer metric on
Ham
(
M
,
ω
)
\textrm {Ham}(M,\omega )
does not extend to a bi-invariant metric on
Symp
0
(
M
,
ω
)
\textrm {Symp}_0(M,\omega )
for many symplectic manifolds. We also show that for the torus
T
2
n
\mathbb T^{2n}
with the standard symplectic form
ω
\omega
, no Finsler metric on
Ham
(
T
2
n
,
ω
)
\textrm {Ham}(\mathbb T^{2n},\omega )
that satisfies a strong form of the invariance condition can extend to a bi-invariant metric on
Symp
0
(
T
2
n
,
ω
)
\textrm {Symp}_0(\mathbb T^{2n},\omega )
. Another interesting result is that there exists no
C
1
C^1
-continuous bi-invariant metric on
Symp
0
(
T
2
n
,
ω
)
\textrm {Symp}_0(\mathbb T^{2n},\omega )
.