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In this paper, based on the fixed point index theory for a class of $\widetilde{J}$-multivalued maps on absolute neighbourhood retracts, we introduce the notion of index of solvability for a variational inequality on a Riemannian manifold involving a multivalued vector field. We describe the main properties of this topological characteristic and use it to justify the existence of a solution for a variational inequality problem. As application, the problem of optimization of a non-smooth functional on a Hadamard manifold is considered. [ABSTRACT FROM AUTHOR]