Consider a parametric family $p_\theta(x)$ of distributions, with parameter $\theta \in \Theta \subseteq \mathbb R^p$.
If the mapping $\theta \mapsto p_\theta(x)$ is continuously differentiable at $\theta_0 \in \Theta$ for $p_{\theta_0}$-almost all $x$, then the Fisher information at $\theta_0$ is the $p$-by-$p$ psd matrix given by $F(\theta_0) = \mathbb E_{x \sim p_{\theta_0}}[s_{\theta_0}(x)s_{\theta_0}(x)^T]$, where $x \mapsto s_{\theta_0}(x) := \partial_\theta \log(p_\theta(x))|_{\theta=\theta_0} \in \mathbb R^p$ is the score function at $\theta_0$.
Question
What is the geometric interpretation of the trace of $F(\theta_0)$ ?
Observation
If is well-known that $KL(p_{\theta_0 + \Delta \theta}\|p_{\theta_0}) = \frac{1}{2}\Delta \theta^T F(\theta_0)\Delta \theta + o(\|\Delta\theta\|^2)$.