On the structure of RCD spaces with upper curvature bounds

Abstract

We develop a structure theory for RCD spaces with curvature bounded above in Alexandrov sense. In particular, we show that any such space is a topological manifold with boundary whose interior is equal to the set of regular points. Further the set of regular points is a smooth manifold and is geodesically convex. Around regular points there are DC coordinates and the distance is induced by a continuous BV Riemannian metric.

arXiv commentparagraph in the introduction after the main theorem added, 30 pages, comments are welcome

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Cited by in corpus (2)

• Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds
• Mixed curvature almost flat manifolds