The scalar curvature is the trace of the ricci tensor. First variation for the mean curvature of hypersurface. On the other hand it can be seen in analogy to a result of bishop and alexander in the setting of alexandrov spaces with curvature bounded from below alexander and. Ricci curvature lower bounds in riemannian geometry and its consequences to more general metric measure spaces. We construct metrics of positive ricci curvature on some vector bundles over tori or more generally, over nilmanifolds. Ricci curvature and bochner formulas for martingales. These notions are based upon more general concepts developed by r. Ricci curvature for metricmeasure spaces via optimal. Ricci curvature, a logarithmic sobolev inequality holds. Ricci flow with surgery on fourmanifolds with positive isotropic curvature chen, binglong and zhu, xiping, journal of differential geometry, 2006 on the complex structure of kahler manifolds with nonnegative curvature chau, albert and tam, luenfai, journal of differential geometry, 2006. Let be the sectional curvature at the point in the direction of the surface element defined by the vectors and, let be normalized vectors orthogonal to each other and to the vector, and. Ricci curvature is also special that it occurs in the einstein equation and in the ricci ow. These conditions are generalizations of ricci curvature lower bounds.
In this context, most of the interpretations are useless. A vanishing theorem for elliptic genera under a ricci. Pdf combinatorial ricci curvature for image processing. The term \comparison geometry had its origin in connection with the success of the rauch comparison theorem and its more powerful global version. The ricci curvature plays a very important role on geometric analysis on riemannian manifolds. The scalar curvature is the contraction of the ricci tensor, and is written as r without subscripts or arguments r g.
Comparison geometry plays a very important role in the study of manifolds with lower ricci curva. Formanricci curvature is a statistics for graphs, which is based on riemannian geometry, and that stresses the relational character of vertices in a network through. Ricci curvature on m to estimates for bounded ricci curvature on pm, e. Pdf the comparison geometry of ricci curvature shunhui. Pdf discrete ricci curvatures for directed networks. Null space constant curvature bianchi identity symmetric tensor ricci curvature these keywords were added by machine and not by the authors. We extend the concept of singular ricci flow by kleiner and lott from 3d compact manifolds to 3d complete manifolds with possibly unbounded curvature. At a high level, financial systems may be represented as weighted graphs that characterize the complex web of interacting agents and information flow for example, debt, stock returns, and shareholder. Ollivier ricci curvature for general graph laplacians. Pdf on the sum of riccicurvatures for weighted graphs. A geometric understanding of ricci curvature in the. On the one hand our result covers a theorem of bacher and sturm concerning euclidean and spherical cones bacher and sturm.
We derive subriemannian ricci curvature tensor for subriemannian manifolds. Ricci curvature bounds for warped products sciencedirect. We direct the reader to 2 for a historical discussion of the study of the ricci tensor, as well as for a summary of what is now known. Rigidity of manifolds with boundary under a lower ricci curvature bound sakurai, yohei, osaka journal of mathematics, 2017. Pdf the ricci curvature of submanifolds and its applications. On a 2manifold, it is equal to twice the gaussian curvature. Contents 1 basic tools and characterizations of ricci curvature lower bound 5. In this paper we generalize the monotonicity formulas of colding acta math 209. Ricci curvature plays an important role in general relativity, where it is the key term in the einstein field equations. Let m be a compact nmanifold with a metric of kpositive ricci curvature, 2. Ricci curvature for parametric statistics via optimal transport. The ricci tensor is a second order tensor about curvature while the stress. These tools have many applications, see next two chapters.
M either convex or empty, and that the ricci curvature is bounded below byk for some k. Assume that m is a ddimensional connected, complete riemannian manifold with. We refer to the nice survey lott and the references therein for a discussion of the geometric interest of lower bounds on ricci curvature and the need for a generalized notion of positive ricci curvature for metric measure spaces. We consider solutions m,gt, 0 download pdf abstract. Monotone quantities play a key role in analysis and geometry. The ricci curvature can be expressed in terms of the sectional curvatures of. A ricci curvature bound is weaker than a sectional curvature bound but stronger than a scalar curvature bound. A new combinatorial ricci curvature and laplacian opera tors for grayscale images are introduced and tested on 2d medical im ages.
This process is experimental and the keywords may be updated as the learning algorithm improves. The ricci curvature tensor eld r is given by r x r. Pdf we modify the definition of ricci curvature of ollivier of markov chains on graphs to study the properties of the. The metric volume form induced by the metric tensor gis the nform. Many results have been proved on manifolds with ricci.
The trace depends on the metric since the ricci tensor is a 0,2valent tensor. Since these manifolds have special holonomy, one might ask. Ricci curvature and monotonicity for harmonic functions. In fact there are ricci flat manifolds which do not admit positive scalar curvature, e. Given 0 and 0 0 such that, for any m of dimension nwith ricm n.
Pdf ricci curvature and eigenvalue estimate on locally. R often times, partial derivatives will be represented with a comma. Moreover, we are able to obtain the myers type theorem for minimal submanifolds in ambient. Comparison geometry for ricci curvature guofang wei, uc santa barbara august, 2008. Rab rc abc nb there is no widely accepted convention for the sign of the riemann curvature tensor, or the ricci tensor, so check the sign conventions of whatever book you are reading. Heat equation, laplacian comparison, nonexplosion and diameter bounds. The ricci operator of an oriented closed riemannian manifold can be realized as a spectral. Following the seminal ideas of lottstrumvillani, we define this notion based on the geodesic convexity of the kullbackleibler divergence in a wasserstein statistical manifold, that is, a manifold of probability distributions endowed with a wasserstein metric tensor structure. A vanishing theorem for elliptic genera under a ricci curvature bound. We show that for ndimensional manifolds with ricci curvature bounded between two negative constants the order of their. In riemannian geometry, the natural framework for the study of spaces with positive curvature seems to be a lower bound on ricci curvature see e. As an application of the generalized singular ricci flow, we show that for any 3d complete riemannian manifold with nonnegative ricci curvature, there exists a smooth ricci flow starting from it.
We introduce some new curvature quantities such as conformal ricci curvature and biricci curvature and extend the classical myers theorem under these new curvature conditions. Many results are established on manifolds with nonnegative ricci curvature or on manifolds with ricci curvature bounded below. One reason the ricci curvature is important in geometry is that it can place restrictions on the topology of manifolds. Ricci curvature also appears in the ricci flow equation, where certain oneparameter families of riemannian metrics are singled out as solutions of a geometricallydefined partial differential equation. Ricci curvature of the internet topology computer science, stony. Everything known comes from these, one way or an other. The sectional and ricci curvatures so again the ricci curvature function ricis not a function on m, but a function on the unit sphere bundle sm. The definitions of curvatures that are easier to generalize to a discrete graph setting are sectional curvature and. The scalar curvature s commonly also r, or sc is defined as the trace of the ricci curvature tensor with respect to the metric. Estimate of distances and angles for positive ricci curvature. Some integral curvature estimates for the ricci flow in. We prove generalized lower ricci curvature bounds for warped products over complete finsler manifolds. We elaborate the notion of a ricci curvature lower bound for parametrized statistical models. Ricci curvature on a graph 6,24, 28 has been introduced which is analogous to the notion of ricci curvature in riemannian geometry 4,33.
900 1437 1410 547 583 1132 54 1186 608 1576 1018 753 1429 660 1079 561 899 1019 754 785 363 1129 978 344 558 435 1266 1333 1378 24 928 674 296 1299 307 836 944 152 872