Metric Structures for Riemannian and Non-Riemannian Spaces
@inproceedings{Gromov1999MetricSF, title={Metric Structures for Riemannian and Non-Riemannian Spaces}, author={Mikhael Gromov}, year={1999} }
Length Structures: Path Metric Spaces.- Degree and Dilatation.- Metric Structures on Families of Metric Spaces.- Convergence and Concentration of Metrics and Measures.- Loewner Rediscovered.- Manifolds with Bounded Ricci Curvature.- Isoperimetric Inequalities and Amenability.- Morse Theory and Minimal Models.- Pinching and Collapse.
2,576 Citations
Lipschitz and path isometric embeddings of metric spaces
- MathematicsGeometriae Dedicata
- 2012
We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash C1 Embedding Theorem.…
Lipschitz and path isometric embeddings of metric spaces
- Mathematics
- 2010
We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash C1 Embedding Theorem.…
The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces
- Mathematics
- 2012
Equipped with the L^2-distortion distance, the space "X" of all metric measure spaces (X,d,m) is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are…
A Course in Metric Geometry
- Mathematics
- 2001
Metric Spaces Length Spaces Constructions Spaces of Bounded Curvature Smooth Length Structures Curvature of Riemannian Metrics Space of Metric Spaces Large-scale Geometry Spaces of Curvature Bounded…
On the measure contraction property of metric measure spaces
- Mathematics
- 2007
We introduce a measure contraction property of metric measure spaces which can be regarded as a generalized notion of the lower Ricci curvature bound on Riemannian manifolds. It is actually…
Convergence of Brownian motions on metric measure spaces under Riemannian Curvature–Dimension conditions
- MathematicsElectronic Journal of Probability
- 2019
We show that the pointed measured Gromov convergence of the underlying spaces implies (or under some condition, is equivalent to) the weak convergence of Brownian motions under Riemannian…
A weakly second-order differential structure on rectifiable metric measure spaces
- Mathematics
- 2014
We give the definition of angles on a Gromov-Hausdorff limit space of a sequence of complete n-dimensional Riemannian manifolds with a lower Ricci curvature bound. We apply this to prove there is a…
Curvature of sub-Riemannian spaces
- Mathematics
- 2003
To any metric spaces there is an associated metric profile. The rectifiability of the metric profile gives a good notion of curvature of a sub-Riemannian space. We shall say that a curvature class is…
Metric conformal structures and hyperbolic dimension
- Mathematics, Computer Science
- 2007
A stereographic projection ofďa is defined and it is shown that it is a metric conformally equivalent to ďa that makes the Isom(X)-action on ∂X bi-Lipschitz, Mobius, symmetric and conformal.
Ricci curvatures in Carnot groups
- Mathematics
- 2013
We study metric contraction properties for metric spaces associated with left-invariant sub-Riemannian metrics on Carnot groups. We show that ideal sub-Riemannian structures on Carnot groups satisfy…