A Riemannian viewpoint on the Amari–Čencov(Formula presented) (Formula presented)-connections and Proudman–Johnson equations

We give a new geometric interpretation of the Amari–Čencov (Formula presented) (Formula presented) -connections (Formula presented) (Formula presented) from information geometry: On the space of densities (Formula presented) (Formula presented), we show that there exist Riemannian metrics (Formula presented) (Formula presented), which we call (Formula presented) (Formula presented) -Fisher–Rao (FR) metrics, whose Levi-Civita connections are (Formula presented) (Formula presented). With the exception of (Formula presented) (Formula presented) (the FR metric), these metrics are non-invariant to the action of the diffeomorphism group (Formula presented) (Formula presented), even though the connections are invariant. This gives a new way of interpreting the geodesics of the (Formula presented) (Formula presented) as energy-minimizing curves. On the space of probability densities (Formula presented) (Formula presented), we show that the same phenomenon holds for (Formula presented) (Formula presented) and that the (Formula presented) (Formula presented) -connections are not metric otherwise. We show that (Formula presented) (Formula presented) -geodesics on this space can be interpreted as radial projections of straight lines on appropriate hyper-surfaces, and use this geometric picture to obtain geodesic convexity for any (Formula presented) (Formula presented). In addition, we prove analogous results for appropriate metrics and connections on (Formula presented) (Formula presented), which, for the case (Formula presented) (Formula presented), imply that the generalized Proudman–Johnson equations on the real line are the Euler–Arnold-type equations of non-right invariant metrics. Finally, in the finite-dimensional case, we show that (Formula presented) (Formula presented) can be metric or non-metric depending on the statistical model considered.