Efficiency for multitime vector variational problems on Riemannian manifolds involving geodesic quasiinvex functionals

We study the connection between a multitime scalar variational problem (SVP), a multitime vector variational problem (VVP) and a multitime vector fractional variational problem (VFP). For (SVP), we establish necessary optimality conditions. For both vector variational problems, we define the notions of Pareto efficient solution and of normal efficient solution and we establish necessary efficiency conditions for (VVP) and (VFP) using both notions. The main purpose of the paper is to establish sufficient efficiency conditions for the vector problems (VVP) and (VFP). Moreover, we obtain sufficient optimality conditions for (SVP). The sufficient conditions are based on our original notion of $(\rho ,b)$-geodesic quasiinvexity.