  Choosing the Fisher information as the metric tensor for a Riemannian
manifold provides a powerful yet fundamental way to understand statistical
distribution families. Distances along this manifold become a compelling
measure of statistical distance, and paths of shorter distance improve sampling
techniques that leverage a sequence of distributions in their operation.
Unfortunately, even for a distribution as generally tractable as the
multivariate normal distribution, this information geometry proves unwieldy
enough that closed-form solutions for shortest-distance paths or their lengths
remain unavailable outside of limited special cases. In this review we present
for general statisticians the most practical aspects of the Fisher geometry for
this fundamental distribution family. Rather than a differential geometric
treatment, we use an intuitive understanding of the covariance-induced
curvature of this manifold to unify the special cases with known closed-form
solution and review approximate solutions for the general case. We also use the
multivariate normal information geometry to better understand the paths or
distances commonly used in statistics (annealing, Wasserstein). Given the
unavailability of a general solution, we also discuss the methods used for
numerically obtaining geodesics in the space of multivariate normals,
identifying remaining challenges and suggesting methodological improvements.
