The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 616–655 | Cite as

Sectional Curvature-Type Conditions on Metric Spaces

  • Martin KellEmail author


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 Poincaré 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.


Sectional curvature Non-Riemannian metric space Splitting theorem Soul theorem 

Mathematics Subject Classification

53C23 (51F99) 


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Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany

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