I have a sequence of symmetric positive definite matrices in $GL(n)$ (in my case, covariance matrices of some gaussians) and vectors $R^n$ (the mean of these gaussians). I consider this sequence to be the discretization of a curve embedded in $GL(n)\times R^n$.

I would like to compute the embedding curvature of this curve at each of my sample points, using the Wasserstein geometry (see Wasserstein Geometry of Gaussian Measures), not information geometry (like The Riemannian Geometry of the Space of Positive-Definite Matrices ...).

I have some difficulties understanding the parameterization of the space of SPD matrices in the Wasserstein Geometry article above and thus, I am not able to compute the Levi-Civita connection and its projection on my curve with respect to the metric. Could anyone help me understand that (or anything else you feel I might not have understood - I feel I barely understand these things) ? Does anyone know of relevant work ?

Thanks!