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On the structure of RCD spaces with upper curvature bounds
TLDR
In this article, a structure theory for RCD spaces with curvature bounded above in Alexandrov sense is developed, and it is shown that any such space is a topological manifold with boundary whose interior is equal to the set of regular points.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.read more
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Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds
TL;DR: In this article, the authors investigated Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary and proved several measure rigidity results for some important functional and geometric inequalities.
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Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds
TL;DR: In this paper, the authors investigated Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary and proved measure rigidity results related to optimal transport.
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Stability of metric measure spaces with integral Ricci curvature bounds
TL;DR: In this article, it was shown that a sequence of n-dimensional Riemannian manifolds subconverges to a metric measure space that satisfies the curvature-dimension condition C D ( K, n ) in the sense of Lott-Sturm-Villani provided the L p -norm for p > n 2 of the part of the Ricci curvature that lies below K converges to 0.
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Remarks on Manifolds with Two-Sided Curvature Bounds
TL;DR: In this article, the authors discuss folklore statements about distance functions in manifolds with two-sided bounded curvature, including regularity, subsets of positive reach and the cut locus.
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Mixed curvature almost flat manifolds
TL;DR: In this article, a mixed curvature analogue of Gromov's almost flat manifolds theorem for upper sectional and lower Bakry-Emery Ricci curvature bounds is presented.
References
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Transport maps, non-branching sets of geodesics and measure rigidity
TL;DR: In this paper, the authors investigated the relationship between a general existence of transport maps of optimal couplings with absolutely continuous first marginal and the property of the background measure called essentially non-branching introduced by Rajala-Sturm.
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Diameter rigidity of spherical polyhedra
Werner Ballmann,Michael Brin +1 more
TL;DR: In this paper, the rank rigidity of spherical polyhedra of non-positive curvature is investigated. And the rank of a complete, simply connected space Y of nonpositive curvatures is greater than or equal to 2 if every geodesic segment in Y is contained in an isometrically embedded, convex Euclidean plane.
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A Sufficient Condition to a Regular Set Being of Positive Measure on Spaces
TL;DR: In this paper, the authors studied regular sets in metric measure spaces with Ricci curvature bounded from below and proved that the existence of a point in the regular set of the highest dimension implies the positivity of the measure of such regular set.
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Affine functions on CAT (κ)-spaces
TL;DR: In this article, affine functions on spaces with an upper curvature bound are described, and affine function functions on affine spaces with upper curvatures are described for affine networks.