  The Wasserstein distance on multivariate non-degenerate Gaussian densities is
a Riemannian distance. After reviewing the properties of the distance and the
metric geodesic, we present an explicit form of the Riemannian metrics on
positive-definite matrices and compute its tensor form with respect to the
trace inner product. The tensor is a matrix which is the solution to a Lyapunov
equation. We compute the explicit formula for the Riemannian exponential, the
normal coordinates charts and the Riemannian gradient. Finally, the Levi-Civita
covariant derivative is computed in matrix form together with the differential
equation for the parallel transport. While all computations are given in matrix
form, nonetheless we discuss also the use of a special moving frame.
