Monotonicity of the Complementary Vector Field of a Nonexpansive Map

The Minty-Browder monotonicity notion will be generalized for vector fields of a Riemannian manifold M. If M is a Hadamard manifold, complementary vector fields of maps f : M M will be introduced. If f is nonexpansive it is proved that the complementary vector field of f is monotone. In particular, compositions of projection maps onto convex sets will be considered.