Let $V$ be a finite-dimensional vector space. Let $K \subset V$ be a proper cone. That is,
- $\alpha K \subset K$ for $\alpha \geq 0$ (cone)
- $K$ is closed
- $K$ is convex
- $K \cap (-K) = \emptyset$ (pointed)
- $\text{int}(K) \not = \emptyset$ (solid)
I'm looking for ways to equip $\text{int}(K)$ with a Riemannian metric so that it is a Hadamard manifold. That is $K$ with the metric is geodesically complete and has everywhere negative sectional curvature. This is reminiscent of the affine-invariant Riemannian metric often equipped on the manifold of positive-definite matrices.
I was wondering if there's a standard way of doing this, a way with lots of nice properties. I would appreciate any direction in literature.