The $L^p$-geometry and its applications

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.