  Wasserstein distance, especially among symmetric positive-definite matrices,
has broad and deep influences on development of artificial intelligence (AI)
and other branches of computer science. A natural idea is to describe the
geometry of $SPD\left(n\right)$ as a Riemannian manifold endowed with the
Wasserstein metric. In this paper, by involving the fiber bundle, we obtain
explicit expressions for some locally geometric quantities, including
geodesics, exponential maps, the Riemannian connection, Jacobi fields and
curvatures. Furthermore, we discuss the behaviour of geodesics and prove that
the manifold is globally geodesic convex with non-negative curvatures but no
conjugate pair and cut locus. According to arithmetic estimates, we find
curvatures can be controlled by the minimal eigenvalue.
