The Fisher Metric of the Ricci Flow Heat Kernel

We introduce and study a Fisher information metric $g_{\tau }^F$ associated to the conjugate heat kernel of a Ricci flow $(M^n,g_t)$. This tensor measures, at a fixed scale, how the pointed heat-kernel measure changes when the base point is moved. We prove that $g_{\tau }^F$ is monotone in scale and satisfies $g_{\tau }^F\le g_t$. We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect $g_t-g_{\tau }^F\mathrm{.}$ This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities $0<g_{\tau }^F<g_t$ at every positive scale, while in the complete case equality forces a Euclidean splitting. We also develop several consequences of this point of view. These include a sharp reverse Poincaré inequality for the heat semigroup, a contraction formula for $\phi$-divergences along conjugate heat flow, and a canonical construction of heat-kernel splitting maps from large eigenvalues of the Fisher endomorphism. As applications, we relate pointed Nash entropy close to $0$ to small Fisher deficit at comparable scales, and we obtain a codimension-one Fisher-metric criterion for applying Bamler’s $\epsilon$-regularity theorem.