  We study the meromorphic open-string vertex algebras and their modules over
the two-dimensional Riemannian manifolds that are complete, connected,
orientable, and of constant sectional curvature $K\neq 0$. Using the parallel
tensors, we explicitly determine a basis for the meromorphic open-string vertex
algebra, its modules generated by eigenfunctions of the Laplace-Beltrami
operator, and their irreducible quotients. We also study the modules generated
by lowest weight subspace satisfying a geometrically interesting condition. It
is showed that every irreducible module of this type is generated by some
(local) eigenfunction on the manifold. A classification is given for modules of
this type admitting a composition series of finite length. In particular and
remarkably, if every composition factor is generated by eigenfunctions of
eigenvalue $p(p-1)K$ for some $p\in \mathbb{Z}_+$, then the module is
completely reducible.
