The central question in optimal transport is the following: how does one "optimally" move a unit mass from one distribution to another? A computationally feasible way to approximately compute this optimal transport plan is done by regularizing with a quantity called relative entropy, giving rise to an object called the Schrödinger bridge. In this talk, we present a collection of explicit diffusion approximations to small temperature (i.e. small regularization parameter) Schrödinger bridges. In the case when both marginals are the same, the setting we consider includes manifold-valued diffusions and Schrödinger bridges computed with respect to reversible reference diffusions. By choosing the underlying space to be a Hessian manifold, this approach gives an explicit diffusion approximation to the Euclidean Schrödinger bridge with different marginals. Joint work with Soumik Pal.