Characterization of Convex and Generalized Convex Vector Fields on Riemannian Manifolds

In this paper, we define the concepts of C-convexity and generalized C-convexity of vector fields on Riemannian manifolds and we prove that a locally bounded C-convex vector field on Riemannian manifolds is locally Lipschitz. A new definition of subdifferential of a C-convex vector field is introduced and some of its properties similar to those in the scalar case are shown. The inclusive relations between Clarke generalized Jacobian and Mordukhovich coderivative and this subdifferential are proved. Moreover, the C-convexity and C-quasiconvexity of a vector field and the C-monotonicity and C-quasimonotonicity of its Mordukhovich coderivative are studied. We also present a second-order characterization of C-convex vector fields on Riemannian manifolds.