Search results for "Splitting theorem"

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Euclidean spaces as weak tangents of infinitesimally Hilbertian metric spaces with Ricci curvature bounded below

2013

We show that in any infinitesimally Hilbertian CD* (K,N)-space at almost every point there exists a Euclidean weak tangent, i.e., there exists a sequence of dilations of the space that converges to Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof follows by considering iterated tangents and the splitting theorem for infinitesimally Hilbertian CD* (0,N)-spaces.

Mathematics - Differential GeometryPure mathematicsGeneral MathematicsSpace (mathematics)01 natural sciencesMeasure (mathematics)Mathematics - Metric Geometry0103 physical sciencesFOS: MathematicsMathematics::Metric Geometry0101 mathematics[MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG]tangent spaces; non-smooth geometryRicci curvatureMathematics51F99-53B99non-smooth geometrySequenceEuclidean spaceApplied MathematicsHilbertian spaces010102 general mathematicstangent spacesTangentMetric Geometry (math.MG)Euclidean spacesDifferential Geometry (math.DG)[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]weak tangentsBounded functionSplitting theorem010307 mathematical physics
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Failure of topological rigidity results for the measure contraction property

2014

We give two examples of metric measure spaces satisfying the measure contraction property MCP(K,N) but having different topological dimensions at different regions of the space. The first one satisfies MCP(0,3) and contains a subset isometric to $\mathbb{R}$, but does not topologically split. The second space satisfies MCP(2,3) and has diameter $\pi$, which is the maximal possible diameter for a space satisfying MCP(N-1,N), but is not a topological spherical suspension. The latter example gives an answer to a question by Ohta.

Mathematics - Differential Geometrymetric measure spacesGeodesicPhysics::Instrumentation and DetectorsQuantitative Biology::Tissues and Organsmeasure contraction propertyMetric Geometry (math.MG)53C23 (Primary) 28A33 49Q20 (Secondary)Ricci curvature lower boundsTopologyPotential theorymaximal diameter theoremnonbranchingRigidity (electromagnetism)Mathematics - Metric GeometryDifferential Geometry (math.DG)splitting theoremFOS: MathematicsSplitting theoremContraction (operator theory)AnalysisMathematicsgeodesics
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