This article has two objectives. Firstly, we use the vector variational-like inequalities
problems to achieve local approximate (weakly) efficient solutions of the vector optimization problem
within the novel field of the Hadamard manifolds. Previously, we introduced the concepts of
generalized approximate geodesic convex functions and illustrated them with examples. We see the
minimum requirements under which critical points, solutions of Stampacchia, and Minty weak
variational-like inequalities and local approximate weakly efficient solutions can be identified,
extending previous results from the literature for linear Euclidean spaces. Secondly, we show
an economical application, again using solutions of the variational problems to identify Stackelberg
equilibrium points on Hadamard manifolds and under geodesic convexity assumptions
