On the Teichmüller space
T
(
R
0
)
T(R_0)
of a hyperbolic Riemann surface
R
0
R_0
, we consider the length spectrum metric
d
L
d_L
, which measures the difference of hyperbolic structures of Riemann surfaces. It is known that if
R
0
R_0
is of finite type, then
d
L
d_L
defines the same topology as that of Teichmüller metric
d
T
d_T
on
T
(
R
0
)
T(R_0)
. In 2003, H. Shiga extended the discussion to the Teichmüller spaces of Riemann surfaces of infinite type and proved that the two metrics define the same topology on
T
(
R
0
)
T(R_0)
if
R
0
R_0
satisfies some geometric condition. After that, Alessandrini-Liu-Papadopoulos-Su proved that for the Riemann surface satisfying Shiga’s condition, the identity map between the two metric spaces is locally bi-Lipschitz.
In this paper, we extend their results; that is, we show that if
R
0
R_0
has bounded geometry, then the identity map
(
T
(
R
0
)
,
d
L
)
→
(
T
(
R
0
)
,
d
T
)
(T(R_0),d_L) \to (T(R_0),d_T)
is locally bi-Lipschitz.