# 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…