Geodesic Convexity Types in Riemannian Manifolds
  The usual notion of set-convexity, valid in the classical Euclidean context, metamorphoses into several distinct convexity types in the more general Riemannian setting. By studying this phenomenon in reverse, we characterize complete manifolds for which certain convexity types are assumed a priori to coincide. 
Octavian Mitrea
math.DG