The culmination of this work was a proof of the geometrization conjecture first proposed by William Thurston in the 1970s, which can be thought of as a classification of compact 3-manifolds. /Encoding 7 0 R p What makes the Ricci curvature (as well as other curvature quantities such as the, classical theorems of Riemannian geometry, Basic introduction to the mathematics of curved spacetime, Foundations of Differential Geometry, Volume 1, "The Topology of Open Manifolds with Nonnegative Ricci Curvature", "Manifolds with A Lower Ricci Curvature Bound", https://en.wikipedia.org/w/index.php?title=Ricci_curvature&oldid=1007350585, Articles with unsourced statements from January 2021, Creative Commons Attribution-ShareAlike License, Najman, Laurent and Romon, Pascal (2017): Modern approaches to discrete curvature, Springer (Cham), Lecture notes in mathematics, This page was last edited on 17 February 2021, at 17:58. i /ProcSet[/PDF/Text/ImageC] /BaseFont/AYGSCV+CMR8 A second notion, Forman's Ricci curvature, is based on topological arguments. ⁡ 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 663.6 885.4 826.4 736.8 /Name/F2 Premiere Classe De Chern Et Courbure De Ricci: Preuve De La Conjecture De Calabi Paperback – January 1, 1978 by Societe Mathematique de France (Author) See all formats and editions Hide other formats and editions. (6.91)). = /Font 24 0 R 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.7 562.5 625 312.5 Given a smooth chart (U, ) one then has functions gij : (U) → ℝ and gij : (U) → ℝ for each i and j between 1 and n which satisfy. >> X Conversely, the Ricci form determines the Ricci tensor by, In local holomorphic coordinates zα, the Ricci form is given by. Similarly, if the Ricci curvature is negative in the direction of a given vector ξ, such a conical region in the manifold will instead have larger volume than it would in Euclidean space. R J. Diff. By contrast, excluding the case of surfaces, negative La première partie de cette thèse traite de résultats valables dans le cas d’espaces polonais quelconques. On note x;y: [0;d(x;y)] !M une g eod esique reliant x et y dans M. Z have smaller volume than the corresponding conical region in Euclidean space, at least provided that . In the pseudo-Riemmannian setting, by contrast, the condition 761.6 272 489.6] << 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 This fact motivates, for instance, the introduction of the Ricci flow equation as a natural extension of the heat equation for the metric. Briefly, positive Ricci curvature of a Riemannian manifold has strong topological consequences, while (for dimension at least 3), negative Ricci curvature has no topological implications. ���A9?#�Þ4vHX��Ә��� Qu�Uu���tEKޖ�J��X�-K��pI��ϝ�� In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. , {\displaystyle g^{ij}R_{ij}.} they would then define Il affecte à chaque point d'une variété riemannienne un simple nombre réel caractérisant la courbure intrinsèque de la variété en ce point λ /Type/Encoding J. Gasqui, Connexions à courbure de Ricci donnée, Math Z. 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 0 0 707.2 571.2 544 544 816 816 272 FFfz�f Le tenseur de Ricci est un tenseur d'ordre 2, obtenu comme la trace du tenseur de courbure complet. i 43). stream courbure de ricci pdf Abstract: We show that a complete Riemannian manifold of dimension with $\Ric\ geq n{-}1$ and its -st eigenvalue close to is both. where X1, ..., Xn and Y1, ..., Yn are the components of X and Y relative to the coordinate vector fields of (U, ). The curvature of this connection is the two form defined by. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. R Flot de Ricci Flot de Ricci a bulles 3-vari´et´es non compactes Richard Hamilton ’82 : t → g(t) sur Mn solution de ∂g ∂t = −2Ricg(t) avec g(0) = g0 donn´ee Si Ricg0 = λg0, g(t) = (1 −2λt)g0 Laurent Bessi`eres Courbure de Ricci : flot et rigidit´e diff´erentielle 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] Yamaguchi – A new version of the differentiable sphere theoremInvent. So one can view the functions Rij as associating to any point p of U a symmetric n × n matrix. R Le tenseur de courbure de Riemann décrit complètement la courbure intrinsèque d’un espace quel que soit son nombre de dimensions. M This matrix-valued map on U is called the Ricci curvature associated to the collection of functions gij. Gasqui, J.: Sur la courbure de Ricci d'une connexion linéaire. endobj R << Alternatively, in a normal coordinate system based at p, at the point p, Near any point p in a Riemannian manifold (M, g), one can define preferred local coordinates, called geodesic normal coordinates. Sylvestre F. L. Gallot (born January 29, 1948 in Bazoches-lès-Bray) is a French mathematician, specializing in differential geometry.He is an emeritus professor at the Institut Fourier of the Université Grenoble Alpes, in the Geometry and Topology section.. Education and career. /Filter[/FlateDecode] Nous proposons une définition d’espace « non-collapsed » avec courbure de Ricci minorée et nous généralisons aux espaces RCD le théorème de convergence du volume de Colding et l’estimation de l’écart de dimension de Cheeger-Colding. | Résumé : Nous présenterons un point de vue nouveau sur la courbure de Ricci en géométrie riemannienne, qui permet entre autres de la généraliser à des espaces discrets et d’étendre ainsi certains théorèmes classiques en courbure positive.On obtient aussi des résultats nouveaux, en particulier pour estimer le trou spectral du laplacien sur une variété riemannienne. As can be seen from the second Bianchi identity, one has. 168 (1979), 167-179. Quoiqu’intuitif, ce résultat est difficile à démontrer en temps continu. j << By taking a divergence, and using the contracted Bianchi identity, one sees that Rm M�� Estimation principale dans le cas riemannien Soit M une vari et e riemannienne compl ete (de dimension n) a courbure de Ricci positive. /Length 61 Note also that the complicated formula defining /Encoding 17 0 R We prove stability results for this inequality. Sci. 7 0 obj 173/circlemultiply/circledivide/circledot/circlecopyrt/openbullet/bullet/equivasymptotic/equivalence/reflexsubset/reflexsuperset/lessequal/greaterequal/precedesequal/followsequal/similar/approxequal/propersubset/propersuperset/lessmuch/greatermuch/precedes/follows/arrowleft/spade] N!�n� ) {\displaystyle Z=0,} 0 >> /Type/Encoding These results, particularly Myers' and Hamilton's, show that positive Ricci curvature has strong topological consequences. de π1(B1(p)) engendré par les lacets de longueur inférieure à 2εet le théorème 0.3 est bien une généralisation du théorème 0.1. That is, it defines for each p in M a (multilinear) map, Define for each p in M the map i R /Encoding 7 0 R 27 0 obj 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 → A notable exception is when the manifold is given a priori as a hypersurface of Euclidean space. Since heat tends to spread through a solid until the body reaches an equilibrium state of constant temperature, if one is given a manifold, the Ricci flow may be hoped to produce an 'equilibrium' Riemannian metric which is Einstein or of constant curvature. Le tenseur de Ricci est défini comme une contraction du tenseur de courbure de Riemann [6] : = ∑ =. Rayon de courbure. {\displaystyle R(X,Y)Z} = Conversely, if the (restricted) holonomy of a 2n-dimensional Riemannian manifold is contained in SU(n), then the manifold is a Ricci-flat Kähler manifold (Kobayashi & Nomizu 1996, IX, §4). This Spring School will consist in two courses given by professors Jürgen Jost and Christian Leonard on discrete Ricci curvature. ∇ b Specifically, in harmonic local coordinates the components satisfy, where as endobj Pincement spectral en courbure de Ricci positive Bertrand, Jerome; Abstract. 1. La courbure de Ricci représente la courbure sectionnelle moyenne de tous les triangles partageant un côté donné; elle détecte la présence locale de matière ou d’énergie dans la théorie d’Einstein. {\displaystyle X,Y\in T_{p}M.} R Y 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 154 (2007), no. P. Berard and D. Meyer proved a Faber-Krahn inequality for domains in compact manifolds with positive Ricci curvature. One common source of the Ricci tensor is that it arises whenever one commutes the covariant derivative with the tensor Laplacian. j /FirstChar 33 500 555.6 527.8 391.7 394.4 388.9 555.6 527.8 722.2 527.8 527.8 444.4 500 1000 500 a Then one can check by a calculation with the chain rule and the product rule that, This shows that the following definition does not depend on the choice of (U, ). , p M @MISC{Veysseire12courburede, author = {Laurent Veysseire}, title = {Courbure de Ricci . | Zbl0223.53033 MR303460 Boundedness criteria for bilinear Fourier multipliers : Bourse de … /FontDescriptor 9 0 R Ric So, provided that n ≥ 3 and The implication is that the Riemann curvature, which is a priori a mapping with vector field inputs and a vector field output, can actually be viewed as a mapping with tangent vector inputs and a tangent vector output. Sylvestre Gallot received his doctorate from Paris Diderot University (Paris 7) with thesis under … in the coordinate approach have an exact parallel in the formulas defining the Levi-Civita connection, and the Riemann curvature via the Levi-Civita connection. La première partie de cette thèse traite de résultats valables dans le cas d’espaces polonais quelconques. ⁡ {\displaystyle \varepsilon } /BaseFont/FJINTT+CMBX12 /Subtype/Type1 }, As can be seen from the Bianchi identities, the Ricci tensor of a Riemannian manifold is symmetric, in the sense that. endobj . b {\displaystyle \lambda .} %PDF-1.2 ε p [CA] Première classe de Chern et courbure de Ricci : preuve de la conjecture de Calabi (Séminaire Palaiseau1978). 2 Download full-text PDF. 277.8 500] This is an excellent book to read, whether you are just starting to network or have been networking for a long time. En géométrie riemannienne, la courbure scalaire (ou scalaire de Ricci) est l'outil le plus simple pour décrire la courbure d'une variété riemannienne. 0 �4�6���p)�j"`k}`7���k����{�KF&Aa��WL�'y��v1�8D��׀s�S=�G�xx�g����?HMJ�:sSE��&��X���.�֘���}�z���%m]����W�cBO��:U��%R�eR� Ricci curvature is now known to have no topological implications; Lohkamp (1994) has shown that any manifold of dimension greater than two admits a complete Riemannian metric of negative Ricci curvature. Zbl0397.35028 [CG] J. Cheeger et D. Gromoll - The splitting theorem for manifolds of non negative Ricci curvature. is symmetric and invertible. on vector fields X, Y, Z. g Y OpenURL . A propos de (ii), noter qu une hypothèse sur la courbure de Ricci est beaucoup plus faible qu une hypothèse sur la courbure sectionnelle puisque, dans la direction d un vecteur unitaire tangent donné, la courbure de Ricci est la moyenne des courbures sectionnelles de tous les 2-plans contenant ce vecteur (multipliée par n - 1). 680.6 777.8 736.1 555.6 722.2 750 750 1027.8 750 750 611.1 277.8 500 277.8 500 277.8 /ProcSet[/PDF/Text/ImageC] /LastChar 196 does not necessarily imply The Ricci tensor can also be generalized to arbitrary affine connections, where it is an invariant that plays an especially important role in the study of projective geometry (geometry associated to unparameterized geodesics) (Nomizu & Sasaki 1994). j Ricci curvature also appears in the Ricci flow equation, where certain one-parameter families of Riemannian metrics are singled out as solutions of a geometrically-defined partial differential equation. = The Ricci form is a closed 2-form. /LastChar 196 161/minus/periodcentered/multiply/asteriskmath/divide/diamondmath/plusminus/minusplus/circleplus/circleminus j These are adapted to the metric so that geodesics through p correspond to straight lines through the origin, in such a manner that the geodesic distance from p corresponds to the Euclidean distance from the origin. La courbure de Ricci grossière d’un processus markovien sur un espace polonais est définie comme un taux de contraction local de la distance de Wasserstein W1 entre les lois du processus partant de deux points distincts. to be what would here be called , >> 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 << In 2007, John Lott, Karl-Theodor Sturm, and Cedric Villani demonstrated decisively that lower bounds on Ricci curvature can be understood entirely in terms of the metric space structure of a Riemannian manifold, together with its volume form. endobj n | Zbl 0316.53036 [G-M 2] S. Gallot, D. Meyer. . Indeed, if ξ is a vector of unit length on a Riemannian n-manifold, then Ric(ξ,ξ) is precisely (n − 1) times the average value of the sectional curvature, taken over all the 2-planes containing ξ. Mouse over to zoom – Click to enlarge. - D'un résultat hilbertien à un principe de comparaison entre spectres. << Paris Sér. {\displaystyle Z} , 2 Y zE��F�囹M���nTm�J��ކ�-�2 �8WA��e��;�$�w8n��_���#�@����F=�M{�|Z�jZ��|&�,��SB�� This is discussed from the perspective of differentiable manifolds in the following subsection, although the underlying content is virtually identical to that of this subsection. However, it is quite an important tensor since it reflects an "orthogonal decomposition" of the Ricci tensor. | Une des retombe es de ce s interactions est la naissance d'une the orie \synthe tique" des espaces me triques mesure sa cour-bure de Ricci minore e, venant comple ter la the orie class ique des espaces me triqes a courbure sectionnelle minore e. Dans ce texte (e galeme nt fourni aux actes du 7, pp. >> Abstract: We show that for n dimensional manifolds whose the Ricci curvature is greater or equal to n-1 and for k in {1,,n+1}, the k-th. Z There is an (n − 2)-dimensional family of such 2-planes, and so only in dimensions 2 and 3 does the Ricci tensor determine the full curvature tensor. SkyeP rated it really liked it Aug 08, Learn More – opens in a new window or tab International postage and import charges paid to Pitney Bowes Inc. R Geom.6 (1971), 119-128. . One can then see that the following are equivalent: In the Riemannian setting, the above orthogonal decomposition shows that tr Z endobj X Sur une feuille de papier, la courbure d’un arc peut se mesurer de deux façons : Imaginez un circuit de moto sur un terrain parfaitement plat, parcouru à une vitesse constante. ) >> The tensor was introduced by Ricci for this reason. The Ricci curvature would then vanish along ξ. 0. by, That is, having fixed Y and Z, then for any basis v1, ..., vn of the vector space TpM, one defines. It is even more remarkable that this cancellation of terms is such that the matrix formula relating Rij to Rij is identical to the matrix formula relating gij to gij. The Ricci curvature of the matrix-valued function given by the matrix product JT(g∘y)J is given by the matrix product JT(R∘y)J, where R denotes the Ricci curvature of g. In mathematics, this property is referred to by saying that the Ricci curvature is a "tensorial quantity", and marks the formula defining Ricci curvature, complicated though it may be, as of outstanding significance in the field of differential geometry. 34 0 obj /Font 28 0 R xڽUMO1��W� [���:Io ⁡ k Suppose that (M, g) is an n-dimensional Riemannian or pseudo-Riemannian manifold, equipped with its Levi-Civita connection ∇. endobj stream 1 << On the structure of spaces with Ricci curvature bounded below. endobj {\displaystyle -\operatorname {tr} (X\mapsto \operatorname {Rm} _{p}(X,Y,Z)).} . 2 In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. 875 531.2 531.2 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 i /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 Z Y 4, 159–161 (French, with English summary).MR 832061 Valera Berestovskii and Conrad Plaut, Uniform universal covers of uniform spaces, Topology Appl. C.R. {\displaystyle R_{ij}} This is quite unexpected since, directly plugging the formula which defines gij into the formula defining Rij, one sees that one will have to consider up to third derivatives of y, arising when the second derivatives in the first four terms of the definition of Rij act upon the components of J. Courbure de Ricci : flot et rigidit´e diff´erentielle Laurent Bessi`eres M´emoire d’Habilitation `a Diriger des Recherches Soutenu le 10 d´ecembre 2010 `a l’Institut Fourier devant un jury compos´e de −G´erard Besson (Institut Fourier) −Michel Boileau (Universit´e de Toulouse) /Subtype/Type1 endstream ε 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 611.8 816 Vu la souplesse des métriques à courbure scalaire positive, la question du contrôle métrique de variétés à courbure scalaire positive est plus subtile. /Encoding 7 0 R On connaÎt l'intérÊt porté sur les liaisons entre courbure de Ricci et géométrie conforme d'une variété riemannienne. Le scalaire de Ricci R ou Ric s'obtient à partir du tenseur de Ricci par la relation générale, appliquée à une surface : g The formulas defining Mostow a démontré que lorsque M et N sont compactes, localement symétriques, à courbure négative ou nulle, si M et N ont même type d’homotopie, << /Name/F1 R for any vector fields X, Y, Z. In the case of two-dimensional manifolds, negativity of the Ricci curvature is synonymous with negativity of the Gaussian curvature, which has very clear topological implications. [3] In physical terms, this property is a manifestation of "general covariance" and is a primary reason that Albert Einstein made use of the formula defining Rij when formulating general relativity. /FirstChar 33 notion de courbure (de Ricci) sur des espaces métriques quelconques, qui permet d’étendre certaines propriétés classiques des variétés de courbure positive, comme la concentration de la mesure. in the introductory section is the same as that in the following section. 17 0 obj Discrete notions of Ricci curvature have been defined on graphs and networks, where they quantify local divergence properties of edges. Geometric inequalities for manifolds with Ricci curvature in the Kato class [ Inégalités géométriques pour des variétés dont la courbure de Ricci est dans la classe de Kato ] Carron, Gilles Annales de l'Institut Fourier, Tome 69 (2019) no. Ricci curvature plays an important role in general relativity, where it is the key term in the Einstein field equations.