  In this paper, a new class of Sobolev spaces with kernel function satisfying
a L\'evy-integrability type condition on compact Riemannian manifolds is
presented. We establish the properties of separability, reflexivity, and
completeness. An embedding result is also proved. As an application, we prove
the existence of solutions for a nonlocal elliptic problem involving the
fractional $p(\cdot, \cdot)$-Laplacian operator. As one of the main tools,
topological degree theory is applied.
