On Projectional Subdifferentials and Projectional Coderivatives with Respect to Smooth Manifolds

We study projectional subdifferentials on smooth manifolds and explore two approaches to conceptualize these objects. The first approach views projectional subdifferentials as differential information associated with the considered function on smooth manifolds. Following the framework established by several mathematicians who have extensively surveyed well-known subdifferentials, we derive analogous results for the projectional ones. Inspired by this perspective, we are going to adopt another approach, that is to consider projectional subdifferentials as intrinsic objects on smooth manifolds, meaning that they coincide with the extended versions of the classical subdifferentials for functions defined on smooth manifolds. As a consequence, we demonstrate that the projectional coderivatives of a set-valued map on smooth manifolds can be regarded as the intrinsic coderivatives of the map. Some problems related to these objects, along with their solutions, are also discussed.