Metric Curvatures Revisited: A Brief Overview

  • Emil Saucan
Part of the Lecture Notes in Mathematics book series (LNM, volume 2184)


We survey metric curvatures, special accent being placed upon the Wald curvature, its relationship with Alexandrov curvature, as well as its application in defining a metric Ricci curvature for PL cell complexes and a metric Ricci flow for PL surfaces. In addition, a simple, metric way of defining curvature for metric measure spaces is proposed.



Research partly supported by Israel Science Foundation Grants 221/07 and 93/11 and by European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no [URI-306706].

Part of this work was done while visiting the Max Planck Institute, Leipzig. Their gracious and warm hospitality, as well as their support are gratefully acknowledged.

The author would like to thank to organizers of 2013 CIRM Meeting on Discrete Curvature for the opportunity they gave him to write this book chapter, and especially Pascal Romon for his attentive guidance and support during the process of writing this presentation, as well as of the short conference proceeding notes.

Thanks are also due to the anonymous reviewer for his attentive, insightful and most helpful corrections and suggestions.


  1. 1.
    Abraham, I., Bartal, Y., Neiman, O.: Embedding metric spaces in their intrinsic dimension. In: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 363–372. Society for Industrial and Applied Mathematics, Philadelphia (2008)Google Scholar
  2. 2.
    Alexander, S.B., Bishop, R.L.: Comparison theorems for curves of bounded geodesic curvature in metric spaces of curvature bounded above. Differ. Geom. Appl 6, 67–86 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Appleboim, E., Saucan, E., Zeevi, Y.Y.: Ricci curvature and flow for image denoising and superesolution. In: Proceedings of EUSIPCO, pp. 2743–2747 (2012)Google Scholar
  4. 4.
    Appleboim, E., Hyams, Y., Krakovski, S., Sageev, C., Saucan, E.: The scale-curvature connection and its application to texture segmentation. Theory Appl. Math. Comput. Sci. 3(1), 38–54 (2013)MATHGoogle Scholar
  5. 5.
    Assouad, P.: Étude d’une dimension métrique liée à la possibilité de plongement dans \(\mathbb{R}^{n}\). C. R. Acad. Sci. Paris 288, 731–734 (1979)MathSciNetMATHGoogle Scholar
  6. 6.
    Assouad, P.: Plongements lipschitziens dans \(\mathbb{R}^{n}\). Bull. Soc. Math. France 111, 429–448 (1983)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bačák, M., Hua, B., Jost, J., Kell, M., Schikorra, A.: A notion of nonpositive curvature for general metric spaces. Differ. Geom. Appl. 38, 22–32 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Berestovskii, V.: Introduction of a Riemannian structure in certain metric spaces. Siberian Math. J. 16, 210–221 (1977)Google Scholar
  9. 9.
    Berestovskii, V.: Spaces with bounded curvature and distance geometry. Siberian Math. J 27(1), 8–19 (1986)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Berger, M.: A Panoramic View of Riemannian Geometry. Springer, Berlin (2003)CrossRefMATHGoogle Scholar
  11. 11.
    Bestvina, M.: Geometric group theory and 3-manifolds hand in hand: the fulfillment of Thurston’s vision. Bull. Am. Math. Soc 51(1), 53–70 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Blumenthal, L.M.: Theory and Applications of Distance Geometry. Claredon Press, Oxford (1953)MATHGoogle Scholar
  13. 13.
    Blumenthal, L., Menger, K.: Studies in Geometry. A Series of Books in Mathematics, vol. XIV, 512 pp. W.H. Freeman and Company, San Francisco (1970)Google Scholar
  14. 14.
    Bonciocat, A.I., Sturm, K.T.: Mass transportation and rough curvature bounds for discrete spaces. J. Funct. Anal. 256(9), 2944–2966 (2009). doi:10.1016/j.jfa.2009.01.029. MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Bourgain, J.: On Lipschitz embedding of finite metric spaces in Hilbert space. Israel J. Math 52(1), 46–52 (1985)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Brehm, U., Kühnel, W.: Smooth approximation of polyhedral surfaces regarding curvatures. Geom. Dedicata 12, 435–461 (1982)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Brooks, R.: Differential Geometry (Lecture Notes). Technion, Haifa (2003)Google Scholar
  18. 18.
    Burago, Y.D., Zalgaller, V.A.: Isometric piecewise linear immersions of two-dimensional manifolds with polyhedral metrics into \(\mathbb{R}^{3}\). St. Petersburg Math. J. 7(3), 369–385 (1996)Google Scholar
  19. 19.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, vol. 33. American Mathematical Society, Providence (2001)MATHGoogle Scholar
  20. 20.
    Cassorla, M.: Approximating compact inner metric spaces by surfaces. Indiana Univ. Math. 41, 505–513 (1992)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Cayley, A.: On a theorem in the geometry of position. Camb. Math. J. 2, 267–271 (1841)Google Scholar
  22. 22.
    Cheeger, J.: Finiteness theorems for Riemannian manifolds. Am. J. Math. 90, 61–74 (1970)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Chow, B.: The Ricci flow on the 2-sphere. J. Differ. Geom. 33(2), 325–334 (1991)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Chow, B., Luo, F.: Combinatorial Ricci flows on surfaces. J. Differ. Geom. 63(1), 97–129 (2003)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Corwin, I., Hoffman, N., Hurder, S., Šešum, V., Xu, Y.: Differential geometry of manifolds with density. Rose Hulman Undergraduate J. Math. 7(1), 1–15 (2006)Google Scholar
  26. 26.
    Forman, R.: Bochner’s method for cell complexes and combinatorial Ricci curvature. Discrete Comput. Geom. 29(3), 323–374 (2003)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Giesen, J.: Curve reconstruction, the traveling salesman problem and Menger’s theorem on length. In: Proceedings of the 15th ACM Symposium on Computational Geometry (SoCG), pp. 207–216. ACM, New York (1999)Google Scholar
  28. 28.
    Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Gilboa, G., Appleboim, E., Saucan, E., Zeevi, Y.Y.: On the role of non-local menger curvature in image processing. In: Proceedings of ICIP 2015, pp. 4337–4341. IEEE (Society) (2015)Google Scholar
  30. 30.
    Goswami, M., Li, S.M., Zhang, J., Gao, J., Saucan, E., Gu, X.D.: Space filling curves for 3d sensor networks with complex topology. In: Proceedings of CCCG 2015, pp. 21–30 (2015)Google Scholar
  31. 31.
    Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Modern Birkhäuser Classics, 3rd edn. Birkhäuser, Basel (2007)Google Scholar
  32. 32.
    Gromov, M., Thurston, W.: Pinching constants for hyperbolic manifolds. Invent. math. 86, 1–12 (1987)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Grove, K., Markvorsen, S.: Curvature, triameter and beyond. Bull. Am. Math. Soc. 27, 261–265 (1992)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Grove, K., Markvorsen, S.: New extremal problems for the Riemannian recognition program via Alexandrov geometry. J. Am. Math. Soc. 8, 1–28 (1995)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Grove, K., Petersen, P.: Bounding homotopy types by geometry. Ann. Math. 128, 195–206 (1988)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Grove, K., Petersen, P., Wu, J.Y.: Bounding homotopy types by geometry. Invent. Math. 99(1), 205–213 (1990)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Gu, X.D., Saucan, E.: Metric ricci curvature for pl manifolds. Geometry 2013(Article ID 694169), 12 pp. doi:10.1155/2013/694169 (2003)Google Scholar
  38. 38.
    Gu, X.D., Yau, S.T.: Computational Conformal Geometry. International Press, Somerville, MA (2008)MATHGoogle Scholar
  39. 39.
    Haas, J.: Personal communication (2013)Google Scholar
  40. 40.
    Hamilton, R.S.: The Ricci flow on surfaces. Contemp. Math. 71, 237–262 (1988)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Han, Q., Hong, J.X.: Isometric embedding of Riemannian manifolds in Euclidean spaces. AMS Math. Surv. 130, 237–262 (2006)MathSciNetMATHGoogle Scholar
  42. 42.
    Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)CrossRefMATHGoogle Scholar
  43. 43.
    Jin, M., Kim, J., Gu, X.D.: Discrete surface Ricci flow: theory and applications. In: IMA International Conference on Mathematics of Surfaces, pp. 209–232. Springer, Berlin (2007)Google Scholar
  44. 44.
    Jin, Y., Jost, J., Wang, G.: A nonlocal version of the Osher-Sole-Vese model. J. Math. Imaging Vision 44(2), 99–113 (2012)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Jin, Y., Jost, J., Wang, G.: A new nonlocal H 1 model for image denoising. J. Math. Imaging Vision 48(1), 93–105 (2014)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Jost, J., Liu, S.: Ollivier’s Ricci curvature, local clustering and curvature dimension inequalities on graphs. Discrete Comput. Geom. 51, 300–322 (2014)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Kay, D.C.: Arc curvature in metric spaces. Geom. Dedicata 9(1), 91–105 (1980)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Kindermann, S., Osher, S., Jones, P.W.: Deblurring and denoising of images by nonlocal functionals. Multiscale Model. Simul. 4(4), 1091–1115 (2005)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Krauthgamer, R., Linial, N., Magen, A.: Metric embeddings – beyond one-dimensional distortion. Discrete Comput. Geom. 31, 339–356 (2004)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Lebedeva, N., Matveev, V., Petrunin, A., Shevchishin, V.: On d-dimensional d-semimetrics and simplex-type inequalities for high-dimensional sine functions. J. Approx. Theory 156(1), 52–81 (2009)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Lebedeva, N., Matveev, V., Petrunin, A., Shevchishin, V.: Smoothing 3-dimensional polyhedral spaces. Electron. Res. Announc. Math. Sci. 22, 12–19 (2015)MathSciNetMATHGoogle Scholar
  52. 52.
    Lin, Y., Lu, L., Yau, S.T.: Ricci curvature of graphs. Tohoku Math. J. 63(4), 605–627 (2011)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Lin, S., Luo, Z., Zang, J., Saucan, E.: Generalized Ricci curvature based sampling and reconstruction of images. In: Proceedings of EUSIPCO 2015, pp. 604–608 (2015)Google Scholar
  54. 54.
    Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some of its algorithmic applications. Combinatorica 15, 215–245 (1995)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Loisel, B., Romon, P.: Ricci curvature on polyhedral surfaces via optimal transportation. Axioms 3(1), 119–139 (2014). CrossRefMATHGoogle Scholar
  56. 56.
    Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (2) 169(3), 903–991 (2009). doi:10.4007/annals.2009.169.903.
  57. 57.
    Luukkainen, J., Saksman, E.: Every complete doubling metric space carries a doubling measure. Proc. Am. Math. Soc. 162(2), 903–991 (2009)MathSciNetMATHGoogle Scholar
  58. 58.
    Mémoli, F.: On the use of Gromov-Hausdorff distances for shape comparison. In: Proceedings of the Point Based Graphics, Prague (2007)Google Scholar
  59. 59.
    Mémoli, F.: A spectral notion of Gromov–Wasserstein distance and related methods. Appl. Comput. Harmon. Anal. 30(3), 363–401 (2011)MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Morgan, F.: Manifolds with density. Not. Am. Math. Soc. 52, 853–858 (2001)MathSciNetMATHGoogle Scholar
  61. 61.
    Munkres, J.R.: Elementary Differential Topology (rev. ed.). Princeton University Press, Princeton, NJ (1966)Google Scholar
  62. 62.
    Naitsat, A., Saucan, E., Zeevi, Y.Y.: Volumetric quasi-conformal mappings. In: Proceedings of GRAPP/VISIGRAPP 2015, pp. 46–57 (2015)Google Scholar
  63. 63.
    Naor, A., Neiman, O.: Assouad’s theorem with dimension independent of the snowflaking. Rev. Mat. Iberoam. 28, 1–21 (2012)MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Nash, J.: \(\mathcal{C}^{1}\) isometric imbeddings. Ann. Math. 60, 383–396 (1954)Google Scholar
  65. 65.
    Nash, J.: The embedding problem for Riemannian manifolds. Ann. Math. 63, 20–63 (1956)MathSciNetCrossRefMATHGoogle Scholar
  66. 66.
    Nikolaev, I.G.: Parallel translation and smoothness of the metric of spaces of bounded curvature. Math. Dokl. 21, 263–265 (1980)MATHGoogle Scholar
  67. 67.
    Nikolaev, I.G.: Parallel displacement of vectors in spaces of Alexandrov two-side-bounded curvature. Siberian Math. J. 24(1), 106–119 (1983)CrossRefMATHGoogle Scholar
  68. 68.
    Nikolaev, I.G.: Smoothness of the metric of spaces with bilaterally bounded curvature in the sense of A. Alexandrov. Siberian Math. J. 24, 247–263 (1983)MathSciNetCrossRefMATHGoogle Scholar
  69. 69.
    Novikov, S.P.: Topology I: General Survey. Springer, Berlin/Heidelberg (1996)CrossRefGoogle Scholar
  70. 70.
    Ollivier, Y.: Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256(3), 810–864 (2009). MathSciNetCrossRefMATHGoogle Scholar
  71. 71.
    Otsu, Y.: On manifolds with small excess. Am. J. Math. 115, 1229–1280 (1993)MathSciNetCrossRefMATHGoogle Scholar
  72. 72.
    Pajot, H.: Analytic capacity, rectificabilility, menger curvature and the cauchy integral. In: Lecture Notes in Mathematics (LNM) 1799. Springer, Berlin (2002)Google Scholar
  73. 73.
    Perelman, G.: Alexandrov’s spaces with curvature bounded from below II. Tech. rep., Leningrad Department of the Steklov Institute of Mathematics (LOMI) (1991)Google Scholar
  74. 74.
    Petersen, P.: Riemannian Geometry. Springer, New York (1998)CrossRefMATHGoogle Scholar
  75. 75.
    Plaut, C.: Almost Riemannian spaces. J. Differ. Geom. 34, 515–537 (1991)MathSciNetCrossRefMATHGoogle Scholar
  76. 76.
    Plaut, C.: A metric characterization of manifolds with boundary. Compos. Math. 81(3), 337–354 (1992)MathSciNetMATHGoogle Scholar
  77. 77.
    Plaut, C.: Metric curvature, convergence, and topological finiteness. Duke Math. J. 66(1), 43–57 (1992)MathSciNetCrossRefMATHGoogle Scholar
  78. 78.
    Plaut, C.: Spaces of Wald-Berestowskii curvature bounded below. J. Geom. Anal. 6(1), 113–134 (1996)MathSciNetCrossRefMATHGoogle Scholar
  79. 79.
    Plaut, C.: Metric spaces of curvature ≥ k. In: Daverman, R.J., Sher, R.B. (eds.) Handbook of Geometric Topology, pp. 819–898. Elsevier, Amsterdam (2002)Google Scholar
  80. 80.
    Richard, T.: Canonical smoothing of compact Alexandrov surfaces via Ricci flow. Tech. rep., Université Paris-Est Créteil (2012). ArXiv:1204.5461Google Scholar
  81. 81.
    Robinson, C.V.: A simple way of computing the Gauss curvature of a surface. Reports of a Mathematical Colloquium (Second Series) 5–6(1), pp. 16–24 (1944)Google Scholar
  82. 82.
    Sageev, M.: Cat(0) cube complexes and groups. In: Geometric Group Theory, IAS/Park City Mathematics Series, vol. 21, pp. 7–53. AMS and IAS/PCMI (2014)Google Scholar
  83. 83.
    Sarkar, R., Yin, X., Gao, J., Luo, F., Gu, X.D.: Greedy routing with guaranteed delivery using Ricci flows. In: Proceedings of IPSN’09, pp. 121–132 (2009)Google Scholar
  84. 84.
    Sarkar, R., Zeng, W., Gao, J., Gu, X.D.: Covering space for in-network sensor data storage. In: Proceedings of the 9th ACM/IEEE International Conference on Information Processing in Sensor Networks, pp. 232–243. ACM, New York (2010)Google Scholar
  85. 85.
    Saucan, E.: Surface triangulation – the metric approach. Tech. rep., Technion (2004). Arxiv:cs.GR/0401023Google Scholar
  86. 86.
    Saucan, E.: Curvature – smooth, piecewise-linear and metric. In: What is Geometry? pp. 237–268. Polimetrica, Milano (2006)Google Scholar
  87. 87.
    Saucan, E.: A simple sampling method for metric measure spaces. preprint (2011). ArXiv:1103.3843v1 [cs.IT]Google Scholar
  88. 88.
    Saucan, E.: On a construction of Burago and Zalgaller. Asian J. Math. 16(5), 587–606 (2012)MathSciNetCrossRefMATHGoogle Scholar
  89. 89.
    Saucan, E.: A metric Ricci flow for surfaces and its applications. Geom. Imaging Comput. 1(2), 259–301 (2014)MathSciNetCrossRefMATHGoogle Scholar
  90. 90.
    Saucan, E.: Metric curvatures and their applications. Geom. Imaging Comput. 2(4), 257–334 (2015)MathSciNetCrossRefMATHGoogle Scholar
  91. 91.
    Saucan, E., Appleboim, E.: Curvature based clustering for dna microarray data analysis. In: Lecture Notes in Computer Science, vol. 3523, pp. 405–412. Springer, New York (2005)Google Scholar
  92. 92.
    Saucan, E., Appleboim, E.: Metric methods in surface triangulation. In: Lecture Notes in Computer Science, vol. 5654, pp. 335–355. Springer, New York (2009)Google Scholar
  93. 93.
    Saucan, E., Appleboim, E., Zeevi, Y.: Sampling and reconstruction of surfaces and higher dimensional manifolds. J. Math. Imaging Vision 30(1), 105–123 (2008)MathSciNetCrossRefGoogle Scholar
  94. 94.
    Saucan, E., Appleboim, E., Zeevi, Y.: Geometric approach to sampling and communication. Sampl. Theory Signal Image Process. 11(1), 1–24 (2012)MathSciNetMATHGoogle Scholar
  95. 95.
    Semmes, S.: Bilipschitz mappings and strong a weights. Acad. Sci. Fenn. Math. 18, 211–248 (1993)MathSciNetMATHGoogle Scholar
  96. 96.
    Semmes, S.: Metric spaces and mappings seen at many scales. In: Metric Structures for Riemannian and non-Riemannian Spaces. Progress in Mathematics, vol. 152, pp. 401–518. Birkhauser, Boston (1999)Google Scholar
  97. 97.
    Semmes, S.: Some Novel Types of Fractal Geometry. Clarendon Press, Oxford (2001)MATHGoogle Scholar
  98. 98.
    Shephard, G.C.: Angle deficiences of convex polytopes. J. Lond. Math. Soc. 43, 325–336 (1968)CrossRefMATHGoogle Scholar
  99. 99.
    Sonn, E., Saucan, E., Appelboim, E., Zeevi, Y.Y.: Ricci flow for image processing. In: Proceedings of IEEEI 2014. IEEEI (2014)Google Scholar
  100. 100.
    Stone, D.A.: Sectional curvatures in piecewise linear manifolds. Bull. Am. Math. Soc. 79(5), 1060–1063 (1973)MathSciNetCrossRefMATHGoogle Scholar
  101. 101.
    Stone, D.A.: A combinatorial analogue of a theorem of Myers. Ill. J. Math. 20(1), 12–21 (1976)MathSciNetMATHGoogle Scholar
  102. 102.
    Stone, D.A.: Correction to my paper: “A combinatorial analogue of a theorem of Myers” (Illinois J. Math. 20(1), 12–21 (1976). Illinois J. Math. 20(3), 551–554 (1976)Google Scholar
  103. 103.
    Stone, D.A.: Geodesics in piecewise linear manifolds. Trans. Am. Math. Soc. 215, 1–44 (1976)MathSciNetCrossRefMATHGoogle Scholar
  104. 104.
    Sturm, K.T.: On the geometry of metric measure spaces. I and II. Acta Math. 196(1), 65–177 (2006). doi:10.1007/s11511-006-0003-7. MathSciNetMATHGoogle Scholar
  105. 105.
    Villani, C.: Optimal transport, Old and new. In: Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009). doi:10.1007/978-3-540-71050-9.
  106. 106.
    Wald, A.: Sur la courbure des surfaces. C. R. Acad. Sci. Paris 201, 918–920 (1935)MATHGoogle Scholar
  107. 107.
    Wald, A.: Begründung einer koordinatenlosen Differentialgeometrie der Flächen. Ergebnisse eines Math. Kolloquiums, 1. Reihe 7, 24–46 (1936)Google Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Electrical Engineering DepartmentTechnionHaifaIsrael
  2. 2.Mathematics DepartmentTechnionHaifaIsrael

Personalised recommendations