Geodesic monotone vector fields

Having in mind the Minty–Browder monotonicity notion, we shall generalize it for vector fields on Riemannian manifolds, defining the geodesic monotone vector fields. The gradients of geodesic convex functions, important in optimization, linear and nonlinear programming on Riemannian manifolds, are geodesic monotone vector fields. The geodesic monotonicity will be related with the first variation of the length of a geodesic. The connection between the existence of closed geodesics and monotone vector fields will also be analyzed. We give a class of strictly monotone vector fields on a simply connected, complete Riemannian manifold with nonpositive sectional curvature, which generalize the notion of position vector fields. The notion of geodesic scalar derivative will be introduced for characterization of geodesic monotone vector fields on such manifolds. The constant sectional curvature case will be analized separately, since it has important peculiarities.