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

Meromorphic open-string vertex algebras and modules over two-dimensional orientable space forms