The $L^p$-geometry and its applications

TL;DR

This paper extends the Fisher information metric to $L^p$-metrics on various geometric spaces, leading to new insights in diffeomorphism groups, symplectic geometry, and Teichmüller theory.

Contribution

It introduces $L^p$-metrics in information geometry and applies them to diverse geometric contexts, including diffeomorphism groups and symplectic forms, with several new theoretical results.

Findings

Geometry of $ ext{Diff}_{- abla}( r)$ similar to universal Teichmüller space

Analysis of the space of symplectic forms on a manifold

Generalization of Gelfand-Fuchs cocycles to higher dimensions

Abstract

We generalize the classical Fisher information metric on statistical models to $L^p$-metrics on various spaces of differential forms or group of diffeomorphisms. Using this new interpretation from information geometry, we derive several new results in geometry on group of diffeomorphisms, symplectic geometry and Teichm\"{u}ller theory. This includes geometry of $\operatorname{Diff}_{-\infty}(\RR)$, similar to that of universal Teichm\"{u}ller space in essence, also a study on the space of all symplectic forms on a symplectic manifold $M$ and a generalization of Gelfand-Fuchs cocycles to higher-dimension. Furthermore, we answer questions in $\alpha$-geometry posed by Gibilisco, and generalize the $L^p$-metrics to geometry on Orlicz spaces.