Gram Matrix Analysis of Finite Distance Spaces in Constant Curvature

We develop the utility of Gram matrix machinery as a tool to treat the geometry of simplices in space forms. A formula relating the determinant of a normalized Gram matrix to the geometry of the simplex it represents is presented. We then apply the tools to leaf spaces, i.e. the set of degree 1 vertices of a metric tree. One main result is that for a given metric space X there exists a constant ₀ < 0, such that X embeds into all hyperbolic spaces of curvature less than ₀, if and only if X is a leaf space.