Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access June 26, 2021

Remarks on Manifolds with Two-Sided Curvature Bounds

  • Vitali Kapovitch EMAIL logo and Alexander Lytchak

Abstract

We discuss folklore statements about distance functions in manifolds with two-sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.

MSC 2010: 53C20; 53C21; 53C23

References

[1] P. Albano, P. Cannarsa, K. T. Nguyen, and C. Sinestrari, Singular gradient flow of the distance function and homotopy equivalence, Math. Ann. 356 (2013), no. 1, 23–43. MR 3038120Search in Google Scholar

[2] L. Ambrosio, N. Gigli, and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. MR 2129498Search in Google Scholar

[3] S. Alexander, V. Kapovitch, and A. Petrunin, Alexandrov geometry, Preprint, https://arxiv.org/abs/1903.08539 (2019).10.1007/978-3-030-05312-3Search in Google Scholar

[4] P. Albano, On the cut locus of closed sets, Nonlinear Anal. 125 (2015), 398–405. MR 337359110.1016/j.na.2015.06.003Search in Google Scholar

[5] B. Andrews, Notes on the isometric embedding problem and the Nash-Moser implicit function theorem, Surveys in analysis and operator theory (Canberra, 2001), Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 40, Austral. Nat. Univ., Canberra, 2002, pp. 157–208. MR 1953483Search in Google Scholar

[6] V. Bangert, Sets with positive reach, Arch. Math. (Basel) 38 (1982), no. 1, 54–57. MR 646321Search in Google Scholar

[7] Y. Burago, M. Gromov, and G. Perelman, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3–51, 222. MR 1185284Search in Google Scholar

[8] V. N. Berestovskij and I. G. Nikolaev, Multidimensional generalized Riemannian spaces, Geometry, IV, Encyclopaedia Math. Sci., vol. 70, Springer, Berlin, 1993, pp. 165–243, 245–250. MR 126396510.1007/978-3-662-02897-1_2Search in Google Scholar

[9] P. Cannarsa and W. Cheng, Singularities of solutions of Hamilton-Jacobi equations,a rxiv.org/abs/2101.02075, 2021.Search in Google Scholar

[10] P. Cannarsa, W. Cheng, and A. Fathi, Singularities of solutions of time dependent Hamilton-Jacobi equations. applications to Riemannian geometry, arxiv.org/abs/1912.04863, 2019.Search in Google Scholar

[11] P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control, Progress in Nonlinear Differential Equations and their Applications, vol. 58, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 204161710.1007/b138356Search in Google Scholar

[12] D. M. DeTurck and J. L. Kazdan, Some regularity theorems in Riemannian geometry, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 249–260. MR 644518Search in Google Scholar

[13] H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325Search in Google Scholar

[14] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, english ed., Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007, Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates. MR 2307192Search in Google Scholar

[15] M. Ghomi and J. Sprouck, Total curvature and the isoperimetric inequality in Cartan–Hadamard manifolds, Preprint, https://arxiv.org/abs/1908.09814 (2019).Search in Google Scholar

[16] V. Kapovitch, M. Kell, and C. Ketterer, On the structure of RCD spaces with upper curvature bounds, arXiv:1908.07036, 2019.Search in Google Scholar

[17] V. Kapovitch and A. Lytchak, Structure of submetries, Preprint, https://arxiv.org/abs/2007.01325 (2020).Search in Google Scholar

[18] N. Kleinjohann, Nächste Punkte in der Riemannschen Geometrie, Math. Z. 176 (1981), no. 3, 327–344. MR 610214Search in Google Scholar

[19] A. Lytchak and K. Nagano, Geodesically complete spaces with an upper curvature bound, arXiv:1804.05189, 2018.Search in Google Scholar

[20] N. Lebedeva and A. Nepechiy, Alexandrov regions, Preprint (2020).Search in Google Scholar

[21] A. Lytchak and S. Wenger, Isoperimetric characterization of upper curvature bounds, Acta Math. 221 (2018).10.4310/ACTA.2018.v221.n1.a5Search in Google Scholar

[22] A. Lytchak and A. Yaman, On Hölder continuous Riemannian and Finsler metrics, Trans. Amer. Math. Soc. 358 (2006), no. 7, 2917–2926. MR 2216252Search in Google Scholar

[23] A. Lytchak, On the geometry of subsets of positive reach, Manuscripta Math. 115 (2004), no. 2, 199–205. MR 2098470Search in Google Scholar

[24] A. Lytchak, Open map theorem for metric spaces, Algebra i Analiz 17 (2005), no. 3, 139–159. MR 2167848Search in Google Scholar

[25] A. Lytchak, Almost convex subsets, Geom. Dedicata 115 (2005), 201–218. MR 218004810.1007/s10711-005-5994-2Search in Google Scholar

[26] C. Mantegazza and A. C. Mennucci, Hamilton-Jacobi equations and distance functions on Riemannian manifolds, Appl. Math. Optim. 47 (2003), no. 1, 1–25. MR 1941909Search in Google Scholar

[27] I. G. Nikolaev, The Synge’s formula for geodesic variations in a space of bounded curvature by A. D. Aleksandrov, preprint (Russian), 1988.Search in Google Scholar

[28], Bounded curvature closure of the set of compact Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.) 24 (1991), no. 1, 171–177. MR 1056559Search in Google Scholar

[29] Y. Otsu and T. Shioya, The Riemannian structure of Alexandrov spaces, J. Differential Geom. 39 (1994), no. 3, 629–658. MR 1274133Search in Google Scholar

[30] G. Perelman, DC-structures on Alexandrov spaces, preprint, preliminary version (1994).Search in Google Scholar

[31] S. Peters, Convergence of Riemannian manifolds, Compositio Math. 62 (1987), no. 1, 3–16. MR 892147Search in Google Scholar

[32] A. Petrunin, Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal. 8 (1998), no. 1, 123–148. MR 1601854Search in Google Scholar

[33] A. Petrunin, Semiconcave functions in Alexandrov’s geometry, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 137–201. MR 240826610.4310/SDG.2006.v11.n1.a6Search in Google Scholar

[34] J. Rataj and L. Zají£ek, On the structure of sets with positive reach, Math. Nachr. 290 (2017), no. 11-12, 1806–1829. MR 3683461Search in Google Scholar

[35] I. K. Sabitov, On the smoothness of isometries, Sibirsk. Mat. Zh. 34 (1993), no. 4, 169–176, iv, x. MR 1248802Search in Google Scholar

[36] W.-X. Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989), no. 1, 223–301.Search in Google Scholar

[37] I. H. Sabitov and S. Z. Šefel, Connections between the order of smoothness of a surface and that of its metric, Sibirsk. Mat. Ž. 17 (1976), no. 4, 916–925. MR 0425855Search in Google Scholar

[38] M. E. Taylor, Tools for PDE, Mathematical Surveys and Monographs, vol. 81, American Mathematical Society, Providence, RI, 2000, Pseudodifferential operators, paradifferential operators, and layer potentials. MR 1766415Search in Google Scholar

[39] M. Taylor, Existence and regularity of isometries, Trans. Amer. Math. Soc. 358 (2006), no. 6, 2415–2423. MR 2204038Search in Google Scholar

Received: 2021-01-21
Accepted: 2021-05-03
Published Online: 2021-06-26

© 2021 Vitali Kapovitch et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 29.3.2024 from https://www.degruyter.com/document/doi/10.1515/agms-2020-0122/html
Scroll to top button