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 $K eq 0$. 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 $pin mathbb{Z}_+$, then the module is completely reducible.
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