Thus if a cone emitted with an initially circular or spherical cross-section becomes distorted into an ellipse ellipsoid , it is possible for the volume distortion to vanish if the distortions along the principal axes counteract one another. In physical applications, the presence of a nonvanishing sectional curvature does not necessarily indicate the presence of any mass locally; if an initially circular cross-section of a cone of worldlines later becomes elliptical, without changing its volume, then this is due to tidal effects from a mass at some other location. Applications[ edit ] Ricci curvature plays an important role in general relativity , where it is the key term in the Einstein field equations. Ricci curvature also appears in the Ricci flow equation, where a time-dependent Riemannian metric is deformed in the direction of minus its Ricci curvature. This system of partial differential equations is a non-linear analog of the heat equation , and was first introduced by Richard S. Hamilton in the early s.
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The coarse Ricci curvature of a Markov process on a Polish space is defined as a local contraction rate of the W1 Wasserstein distance between the laws of the process starting at two different points.
The first part of this thesis deals with results holding in the case of general Polish spaces. The simplest of them is that the infimum of the coarse Ricci curvature is a global contraction rate of the semigroup of the process for the W1 distance between probability measures.
Though intuitive, this result is diffucult to prove in continuous time. The proof of this result, and the following consequences for the spectral gap of the generator are the subject of Chapter 1.
Another interesting result, using the values of the coarse Ricci curvature at different points, and not only its infimum, is a concentration result for the equilibrium measures, only holding in a discrete time framework. That will be the topic of Chapter 2. The second part of this thesis deals with the particular case of diffusions on Riemannian manifolds.
A formula is given, allowing to get the coarse Ricci curvature from the generator of the diffusion. In the case when the metric is adapted to the diffusion, we show the existence of a coupling between the paths starting at two different points, such that the coarse Ricci curvature is exactly the decreasing rate of the distance between these paths. We can then show that the spectral gap of the generator is at least the harmonic mean of the Ricci curvature.
This result can be generalized when the metric is not the one induced by the generator, but it needs a very restricting hypothesis, and the curvature we have to choose is smaller.
COURBURE DE RICCI PDF
Memi Polar factorization of maps on Riemannian manifolds. Journals Seminars Books Theses Authors. Fukaya — Theory of convergence for Riemannian orbifoldsJapan. DeturckDeforming metrics in the direction of their Ricci tensorsJ. Colding — Large manifolds with positive Ricci curvatureInvent. The variational formulation of the Fokker-Planck equation.
Wu — Geometric finiteness theorems via controlled topologyInv. YauComplete 3-dimensional manifolds with positive Ricci curvature and scalar curvaturein Seminar on Differential Geometry, ed. Ivanov — Courbire asymptotic volume of Tor i, Geom. EhrlichMetric deformations of curvature II: Polar factorization and monotone rearrangement of vector-valued functions.