We discuss the asymptotic stability of the wave equation on a compact Riemannian manifold
(
M
,
g
)
(M, \bf g)
subject to locally distributed viscoelastic effects on a subset
ω
⊂
M
\omega \subset M
. Assuming that the well-known geometric control condition
(
ω
,
T
0
)
(\omega , T_0)
holds and supposing that the relaxation function is bounded by a function that decays exponentially to zero, we show that the solutions of the corresponding partial viscoelastic model decay exponentially to zero. We give a new geometric proof extending the prior results in the literature from the Euclidean setting to compact Riemannian manifolds (with or without boundary).