Sectional Curvature-Type Conditions on Metric Spaces

Abstract

In the first part Busemann concavity as non-negative curvature is introduced and a bi-Lipschitz splitting theorem is shown. Furthermore, if the Hausdorff measure of a Busemann concave space is non-trivial then the space is doubling and satisfies a Poincare condition and the measure contraction property. Using a comparison geometry variant for general lower curvature bounds $$k\in {\mathbb {R}}$$ , a Bonnet–Myers theorem can be proven for spaces with lower curvature bound $$k>0$$ . In the second part the notion of uniform smoothness known from the theory of Banach spaces is applied to metric spaces. It is shown that Busemann functions are (quasi-)convex. This implies the existence of a weak soul. In the end properties are developed to further dissect the soul.