Computational Information Geometry Wonderland

A paper in 1999 by Yoshizawa and Tanabe gave details of multivarite normal geometries: both in the dually flat shape (for KL divergence), and in torsion differential geometry with a general definition of divergences, generalizing the previous Riemannian metric of Skovgaard.

Here is the paper:

Dual dierential geometry associated with the Kullback-Leibler ...

You'll interestingly find the dual divergence and Legendre transformations for the multivariate normal.