There are several notions of entropy for flows arising from geometry, including volume entropy, metric entropy with respect to a natural volume, and topological entropy. General theory shows that metric entropy is always at most the topological entropy. Due to a classical work of Katok, equality occurs only for geodesic flows on negatively cured surfaces only when the underlying manifold is hyperbolic. In this talk, I will explain recent and ongoing work extending this to a general dynamical setting. Joint with J. De Simoi, Martin Leguil and Yun Yang.