  This work presents an explicit description of the Fisher-Rao Riemannian
metric on the Hilbert manifold of equivalent centered Gaussian measures on an
infinite-dimensional Hilbert space. We show that the corresponding quantities
from the finite-dimensional setting of Gaussian densities on Euclidean space,
including the Riemannian metric, Levi-Civita connection, curvature, geodesic
curve, and Riemannian distance, when properly formulated, directly generalize
to this setting. Furthermore, we discuss the connection with the Riemannian
geometry of positive definite unitized Hilbert-Schmidt operators on Hilbert
space, which can be viewed as a regularized version of the current setting.
