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Distributions of distances and volumes of balls in homogeneous lens spaces. (English) Zbl 1482.53066

Summary: Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-dimensional spheres by a free linear action of a finite cyclic group, allows a deeper analysis of their structure. In this paper, we consider the problem of moments for the distance function between randomly selected pairs of points on homogeneous three-dimensional lens spaces. We give a derivation of a recursion relation for the moments, a formula for the \(k\) th moment, and a formula for the moment generating function, as well as an explicit formula for the volume of balls of all radii in these lens spaces.

MSC:

53C30 Differential geometry of homogeneous manifolds
51N25 Analytic geometry with other transformation groups
60D05 Geometric probability and stochastic geometry
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[1] Alday, Luis; Fluder, Martin; Sparks, James, The large N limit of M2-branes on lens spaces, J. High Energy Phys., 2012, Article 57 pp. (2012)
[2] Alexander, James Waddell, Note on two three-dimensional manifolds with the same group, Trans. Am. Math. Soc., 20, 4, 339-342 (1919) · JFM 47.0964.01
[3] Aurich, Ralf; Lustig, Sven, A survey of lens spaces and large-scale cosmic microwave background anisotropy, Mon. Not. R. Astron. Soc., 424, 2, 1556-1562 (2012)
[4] Balch, Brenden; Peterson, Chris; Shonkwiler, Clayton, Expected distances on manifolds of partially oriented flags (2020), preprint
[5] Berrendero, José R.; Cuevas, Antonio; Pateiro-López, Beatriz, Shape classification based on interpoint distance distributions, J. Multivar. Anal., 146, 237-247 (2016) · Zbl 1336.62171
[6] Boileau, Michel, Thick/thin decomposition of three-manifolds and the geometrisation conjecture, (Benedetti, Riccardo; Mantegazza, Carlo, Ricci Flow and Geometric Applications. Ricci Flow and Geometric Applications, Cetraro, Italy 2010. Ricci Flow and Geometric Applications. Ricci Flow and Geometric Applications, Cetraro, Italy 2010, Lecture Notes in Mathematics, vol. 2166 (2016), Springer: Springer Cham), 21-70, Chapter 2 · Zbl 1406.53069
[7] Bonetti, Marco; Pagano, Marcello, The interpoint distance distribution as a descriptor of point patterns, with an application to spatial disease clustering, Stat. Med., 24, 5, 753-773 (2005)
[8] Boutin, Mireille; Kemper, Gregor, On reconstructing n-point configurations from the distribution of distances or areas, Adv. Appl. Math., 32, 4, 709-735 (2004) · Zbl 1072.68103
[9] Brinkman, Daniel; Olver, Peter J., Invariant histograms, Am. Math. Mon., 119, 1, 4-24 (2012) · Zbl 1266.53003
[10] Brody, Elmer Julian, The topological classification of the lens spaces, Ann. Math. (2), 71, 1, 163-184 (1960) · Zbl 0119.18901
[11] Edelman, Alan; Arias, Tomás A.; Smith, Steven T., The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20, 2, 303-353 (1998) · Zbl 0928.65050
[12] William, Anthony, Fairbank Edwards. Gilbert’s sine distribution, Teach. Stat., 22, 3, 70-71 (2000)
[13] Gilbert, Grove Karl, The Moon’s Face: A Study of the Origin of Its Features, Bulletin of the Philosophical Society of Washington, vol. 2, 241-292 (1893)
[14] Gradshteyn, Izrail S.; Ryzhik, Iosif M., Table of Integrals, Series, and Products (2015), Elsevier: Elsevier Amsterdam
[15] Gray, Alfred; Vanhecke, Lieven, Riemannian geometry as determined by the volumes of small geodesic balls, Acta Math., 142, 3-4, 157-198 (1979) · Zbl 0428.53017
[16] Ikeda, Akira; Yamamoto, Yoshihiko, On the spectra of 3-dimensional lens spaces, Osaka J. Math., 16, 2, 447-469 (1979) · Zbl 0415.58018
[17] Lehoucq, Roland; Uzan, Jean-Philippe; Weeks, Jeffrey, Eigenmodes of lens and prism spaces, Kodai Math. J., 26, 1, 119-136 (2003) · Zbl 1044.58041
[18] Facundo, Mémoli, Gromov-Wasserstein distances and the metric approach to object matching, Found. Comput. Math., 11, 4, 417-487 (2011) · Zbl 1244.68078
[19] Mémoli, Facundo; Needham, Tom, Gromov-Monge quasi-metrics and distance distributions (2018), preprint
[20] Osada, Robert; Funkhouser, Thomas; Chazelle, Bernard; Dobkin, David, Shape distributions, ACM Trans. Graph., 21, 4, 807-832 (2002) · Zbl 1331.68256
[21] Polanco, Luis; Perea, Jose A., Coordinatizing data with lens spaces and persistent cohomology, (Friggstad, Zachary; De Carufel, Jean-Lou, Proceedings of the 31st Canadian Conference on Computational Geometry (CCCG 2019) (2019)), 49-58
[22] Przytycki, Józef H.; Yasukhara, Akira, Symmetry of links and classification of lens spaces, Geom. Dedic., 98, 1, 5-61 (2003) · Zbl 1028.57018
[23] Reidemeister, Kurt, Homotopieringe und Linsenräume, Abh. Math. Semin. Univ. Hamb., 11, 1, 102-109 (1935) · Zbl 0011.32404
[24] Salvatore, Paolo; Longoni, Riccardo, Configuration spaces are not homotopy invariant, Topology, 44, 2, 375-380 (2005) · Zbl 1063.55015
[25] Tanaka, Minoru, Compact Riemannian manifolds which are isospectral to three-dimensional lens spaces. II, Proc. Fac. Sci. Tokai Univ., 14, 11-34 (1979) · Zbl 0436.53046
[26] Tietze, Heinrich, Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatshefte Math. Phys., 19, 1-118 (1908) · JFM 39.0171.01
[27] Tomizawa, Shinya, Multicharged black lens, Phys. Rev. D, 100, 2, Article 024056 pp. (2019)
[28] Uzan, Jean-Philippe; Riazuelo, Alain; Lehoucq, Roland; Weeks, Jeffrey, Cosmic microwave background constraints on lens spaces, Phys. Rev. D, 69, 4, Article 043003 pp. (2004)
[29] Viana, Celso, The isoperimetric problem for lens spaces, Math. Ann., 374, 1, 475-497 (2018) · Zbl 1418.53069
[30] Wolf, Joseph A., Spaces of Constant Curvature (2011), AMS Chelsea Publishing: AMS Chelsea Publishing Providence, RI · Zbl 1216.53003
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