Subdifferential theory and the Fenchel conjugate via Busemann functions on Hadamard manifolds

In this paper, we propose a notion of subdifferential defined via Busemann functions and use it to identify a condition under which the Fenchel-Young inequality of Bento, Cruz Neto and Melo (Appl. Math. Optim. 88:83, 2023) holds with equality. This equality condition is particularly significant, as it captures a fundamental duality principle in convex analysis, linking a primal convex function to its conjugate and clarifying the sharpness of the associated inequality on Riemannian manifolds. We also investigate the existence of non-trivial affine functions under Ricci curvature information. In particular, we extend the result of Bento, Cruz Neto and Melo, originally formulated for the case of negative Ricci curvature on an open set, to manifolds whose Ricci curvature may be non-zero. As a consequence, we prove new non-existence criteria for non-trivial affine functions and show that the assumption of non-zero Ricci curvature is, in general, necessary to ensure such a rigidity conclusion.