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Metric Curvatures Revisited: A Brief Overview

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Modern Approaches to Discrete Curvature

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2184))

Abstract

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.

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Notes

  1. 1.

    As the reader will become aware while progressing with this text, we have written previously a book chapters on metric curvatures as well as a largely expository article. However the present paper does not represent a calque of any of these previous ones. For one, the present one is addressed to a much more mathematically literate (not to say “very well educated”) audience than the previous expositions. To be sure, certain repetitions are, unfortunately, unavoidable: After all, the same subject represents the common theme of all these three papers. However, we have strived to keep these at an inevitable minimum. Moreover, we did our best to emphasize different aspects (in general, more modern ones) as well as introducing some novel applications.

  2. 2.

    We know that in mentioning this here we anticipate, somewhat, the reminder of the paper.

  3. 3.

    Although, when these notes were started, the mentioned work was still not published.

  4. 4.

    Named after Haantjes [40], who extended to metric spaces an idea introduced by Finsler in his PhD Thesis.

  5. 5.

    Since it proves us that, indeed, for smooth curves, Haantjes curvature coincides with the classical notion of curvature.

  6. 6.

    Not necessarily geodesic.

  7. 7.

    In any case, it is not truly required and, in fact, even cumbersome in practical applications (see [4, 91] for two such examples).

  8. 8.

    The literature on the subject being too vast to even begin and enumerate it here.

  9. 9.

    As a historical note, it is perhaps worthwhile to recall that Formula (2.14) above was proved by Cayley in his very first mathematical paper [21] (published while he was still begrudgingly making his living as a lawyer!…).

  10. 10.

    Recall that the link lk(v) of a vertex v is the set of all the faces of \(\overline{\mathrm{St}}(v)\) that are not incident to v. Here \(\overline{\mathrm{St}}(v)\) denotes the closed star of v, i.e. the smallest subcomplex (of the given simplicial complex K) that contains St(v), namely \(\overline{\mathrm{St}}(v) =\{\sigma \in \mathrm{ St}(v)\} \cup \{\theta \,\vert \,\theta \leqslant \sigma \}\), where St(v) denotes the star of v, that is the set of all simplices that have v as a face, i.e \(\mathrm{St}(v) =\{\sigma \in K\,\vert \,v\leqslant \sigma \}\).

  11. 11.

    It was, it would appear, Gromov’s observation that, in the geometric setting, the relevant convergence is the Gromov–Hausdorff one.

  12. 12.

    The dimension can be taken as the topological dimension or the Hausdorff dimension—see, e.g. [79].

  13. 13.

    Without getting into the technical subtleties of the definition of the space of directions S p at a point p in a space of bounded curvature, the injectivity radius at p is defined as \(\inf _{\gamma \in S_{p}}\sup _{t}\{\gamma \vert _{[0,t]}\text{is minimal}\}\).

  14. 14.

    This well known “paradox” of the foundations of Geometry is, unfortunately, generally overlooked in certain applications in Imaging and Graphics, which results in a penalty on the quality of the numerical results.

  15. 15.

    Developed avant la lettre.

  16. 16.

    Also, we warn the eventual reader of an unfortunate previously unnoticed typo towards the end of [37].

  17. 17.

    In what would have been probably consider to be a strange—not to say bizarre—development even only a few years ago.

  18. 18.

    Recall that the link Lk(v) of a vertex v is the set of all the faces of \(\overline{\mathrm{St}}(v)\) that are not incident to v. Here \(\overline{\mathrm{St}}(v)\) denotes the closed star of v, i.e. the smallest subcomplex (of the given simplicial complex K) that contains St(v), namely \(\overline{\mathrm{St}}(v) =\{\sigma \in \mathrm{ St}(v)\} \cup \{\theta \,\vert \,\theta \leqslant \sigma \}\), where St(v) denotes the star of v, that is the set of all simplices that have v as a face, i.e \(\mathrm{St}(v) =\{\sigma \in K\,\vert \,v\leqslant \sigma \}\).

  19. 19.

    Note that to apply Richard’s result we have only to consider our surfaces as an Alexandrov surface having curvature bounded from below, condition that is, evidently, satisfied. (In this regard and for a discussion on the definition of Wald/Alexandrov curvature for PL surfaces, see [89, pp. 26–27].

  20. 20.

    For a formal definition and more details see, e.g. [82].

  21. 21.

    Alternatively, this condition may be expressed either as lk(v) “has no missing simplices (Sageev [82]), or as “a nonsimplex contains a non edge” (W. Dicks, see [11]).

  22. 22.

    The author would like to thank the anonymous reviewer for bringing to his attention this paper.

  23. 23.

    Obviously, in the interiors of the faces the metric is already smooth.

  24. 24.

    But not piecewise Euclidean.

  25. 25.

    And, in truth rather trivially, since the result holds, regardless of the specific definition for the curvature of a cell.

  26. 26.

    But, on the other hand, this holds even if n = 3!…

  27. 27.

    Without affecting the analogue of the Bonnet–Myers Theorem—see Sect. 2.2 above.

  28. 28.

    See also [105].

  29. 29.

    When mentioning generalized curvatures for surfaces, one can not fail to mention Morgan’s [60] and his students’ [25] work on “weighted” surfaces and curves.

  30. 30.

    At least, this is the usual convention.

  31. 31.

    However, quasisymmetry represents a much more flexible analytic tool, than the rigid bilipschitz condition—see [42, 96, 97] for a deeper and far more detailed discussion.

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Acknowledgements

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.

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  1. Electrical Engineering Department, Technion, Haifa, 32000, Israel

    Emil Saucan

  2. Mathematics Department, Technion, Haifa, 32000, Israel

    Emil Saucan

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  1. Emil Saucan
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Correspondence to Emil Saucan .

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  1. Laboratoire d’Informatique Gaspard Monge, Université Paris-Est - ESIEE Paris, Noisy-le-Grand, France

    Laurent Najman

  2. Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris-Est Marne-la-Vallée, Marne-la-Vallée, France

    Pascal Romon

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Saucan, E. (2017). Metric Curvatures Revisited: A Brief Overview. In: Najman, L., Romon, P. (eds) Modern Approaches to Discrete Curvature. Lecture Notes in Mathematics, vol 2184. Springer, Cham. https://doi.org/10.1007/978-3-319-58002-9_2

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