The classical Wasserstein-2 distance between Gaussian distributions admits a closed-form expression, giving rise to the Bures–Wasserstein geometry on the cone of positive semi-definite matrices. Analogously, the adapted Wasserstein-2 distance between Gaussian processes also admits a closed-form formula. This naturally endows the space of lower-triangular matrices with a new metric, which we term the adapted Bures–Wasserstein distance. In this talk, we will explore the geometry of this space through the characterization of absolutely continuous curves, geodesics, and tangent cones.