Riemannian manifolds constitute a broad and fruitful framework for the development of different fields in mathematic, such as convex analysis, dynamical systems, optimization or mathematical programming, among other scientific areas, where some of its approaches and methods have successfully been extended from Euclidean spaces. The nonpositive sectional curvature is an important property enjoyed by a large class of differential manifolds, so Hadamard manifolds, which are complete simply connected Riemannian manifolds of nonpositive sectional curvature, have worked out a suitable setting for diverse disciplines.
On the other hand, the study of the class of nonexpansive mappings has become an active research area in nonlinear analysis. This is due to the connection with the geometry of Banach spaces along with the relevance of these mappings in the theory of monotone and accretive operators.
We study the problems that arise in the interface between the fixed point theory for nonexpansive type mappings and the theory of monotone operators in the setting of Hadamard manifolds. Different classes of monotone and accretive set-valued vector fields and the relationship between them will be presented, followed by the study of the existence and approximation of singularities for such vector fields. Then we analyze the problem of finding fixed points of nonexpansive type mappings and the connection with monotonicity. As a consequence, variational inequality and minimization problems in this setting will be discussed.