How does curvature affect KL Divergence?

Question

I have recently started studying Information Geometry and I came across this blog Natural Gradient Descent which talks about the Hessian acting as the curvature of KL divergence. I was wondering,

1. is this curvature analogous to the curvature encountered in Riemannian manifolds, like hyperbolic space has negative curvature?
2. If yes, I wanted to know if there is an intuitive explanation of how changing the curvature value affects the KL divergence?

Any insights are welcome! Thank you in advance.