  In this paper, we introduce a definition of Fenchel conjugate and Fenchel
biconjugate on Hadamard manifolds based on the tangent bundle. Our definition
overcomes the inconvenience that the conjugate depends on the choice of a
certain point on the manifold, as previous definitions required. On the other
hand, this new definition still possesses properties known to hold in the
Euclidean case. It even yields a broader interpretation of the Fenchel
conjugate in the Euclidean case itself. Most prominently, our definition of the
Fenchel conjugate provides a Fenchel-Moreau Theorem for geodesically convex,
proper, lower semicontinuous functions. In addition, this framework allows us
to develop a theory of separation of convex sets on Hadamard manifolds, and a
strict separation theorem is obtained.
