Efficiency for vector variational quotient problems with curvilinear integrals on riemannian manifolds via geodesic quasiinvexity

In the paper, we analyze the necessary efficiency conditions for scalar, vectorial and vector fractional variational problems using curvilinear integrals as objectives and we establish sufficient conditions of efficiency to the above variational problems. The efficiency sufficient conditions use of notions of the geodesic invex set and of (strictly, monotonic) (ρ, b)-geodesic quasiinvex functions.