Wasserstein KL-divergence for Gaussian distributions

We introduce a new version of the KL-divergence for Gaussian distributions which is based on Wasserstein geometry and referred to as WKL-divergence. We show that this version is consistent with the geometry of the sample space ${\mathbb{R}}^n$. In particular, we can evaluate the WKL-divergence of the Dirac measures concentrated in two points which turns out to be proportional to the squared distance between these points.