On the structure of $$\mathrm {RCD}$$ spaces with upper curvature bounds

Abstract

AbstractWe develop a structure theory for $$\mathrm {RCD}$$ 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 $$\mathrm {DC}$$ DC coordinates and the distance is induced by a continuous $$\mathrm {BV}$$ BV Riemannian metric.