# American Institute of Mathematical Sciences

2018, 25: 48-59. doi: 10.3934/era.2018.25.006

## Explicit geodesics in Gromov-Hausdorff space

 Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210. Phone: (614) 292-4975, Fax: (614) 292-1479

Received  March 31, 2017 Revised  March 13, 2018 Published  June 2018

Fund Project: This work was supported by NSF grants CCF-1526513 and IIS-1422400.

We provide an alternative, constructive proof that the collection ${\mathcal{M}}$ of isometry classes of compact metric spaces endowed with the Gromov-Hausdorff distance is a geodesic space. The core of our proof is a construction of explicit geodesics on ${\mathcal{M}}$. We also provide several interesting examples of geodesics on ${\mathcal{M}}$, including a geodesic between ${\mathbb{S}}^0$ and ${\mathbb{S}}^n$ for any $n\geq 1$.

Citation: Samir Chowdhury, Facundo Mémoli. Explicit geodesics in Gromov-Hausdorff space. Electronic Research Announcements, 2018, 25: 48-59. doi: 10.3934/era.2018.25.006
##### References:
 [1] B. Bollobás, The Art of Mathematics: Coffee time in Memphis, Cambridge University Press, New York, 2006. doi: 10.1017/CBO9780511816574. [2] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-12494-9. [3] D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, AMS Graduate Studies in Math., 33, American Mathematical Society, 2001. doi: 10.1090/gsm/033. [4] M. Gromov, Metric Structures for Riemannian and non-Riemannian Spaces, Progress in Mathematics, 152, Birkhäuser Boston Inc., Boston, MA, 1999. [5] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. [6] A. Ivanov, N. Nikolaeva and A. Tuzhilin, The Gromov-Hausdorff metric on the space of compact metric spaces is strictly intrinsic, (Russian) Mat. Zametki, 100 (2016), 947-950; translation in Math. Notes, 100 (2016), 883-885. [7] V. Pestov, Dynamics of Infinite-Dimensional Groups: The Ramsey-Dvoretzky-Milman Phenomenon, University Lecture Series, 40, American Mathematical Soc., Providence, RI, 2006. [8] P. Petersen, Riemannian Geometry, Second edition, Graduate Texts in Mathematics, 171, Springer, New York, 2006. [9] K.-T. Sturm, The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces, preprint, arXiv: 1208.0434, (2012).

show all references

##### References:
 [1] B. Bollobás, The Art of Mathematics: Coffee time in Memphis, Cambridge University Press, New York, 2006. doi: 10.1017/CBO9780511816574. [2] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-12494-9. [3] D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, AMS Graduate Studies in Math., 33, American Mathematical Society, 2001. doi: 10.1090/gsm/033. [4] M. Gromov, Metric Structures for Riemannian and non-Riemannian Spaces, Progress in Mathematics, 152, Birkhäuser Boston Inc., Boston, MA, 1999. [5] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. [6] A. Ivanov, N. Nikolaeva and A. Tuzhilin, The Gromov-Hausdorff metric on the space of compact metric spaces is strictly intrinsic, (Russian) Mat. Zametki, 100 (2016), 947-950; translation in Math. Notes, 100 (2016), 883-885. [7] V. Pestov, Dynamics of Infinite-Dimensional Groups: The Ramsey-Dvoretzky-Milman Phenomenon, University Lecture Series, 40, American Mathematical Soc., Providence, RI, 2006. [8] P. Petersen, Riemannian Geometry, Second edition, Graduate Texts in Mathematics, 171, Springer, New York, 2006. [9] K.-T. Sturm, The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces, preprint, arXiv: 1208.0434, (2012).
Branching geodesics as described in §1.1.2
 [1] Nhan-Phu Chung. Gromov-Hausdorff distances for dynamical systems. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6179-6200. doi: 10.3934/dcds.2020275 [2] Alexanger Arbieto, Carlos Arnoldo Morales Rojas. Topological stability from Gromov-Hausdorff viewpoint. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3531-3544. doi: 10.3934/dcds.2017151 [3] Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320 [4] Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020 [5] Tapio Rajala. Improved geodesics for the reduced curvature-dimension condition in branching metric spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3043-3056. doi: 10.3934/dcds.2013.33.3043 [6] Patrick M. Fitzpatrick, Jacobo Pejsachowicz. Branching and bifurcation. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1955-1975. doi: 10.3934/dcdss.2019127 [7] Ana Cristina Mereu, Marco Antonio Teixeira. Reversibility and branching of periodic orbits. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1177-1199. doi: 10.3934/dcds.2013.33.1177 [8] Dejian Chang, Huili Liu, Jie Xiong. A branching particle system approximation for a class of FBSDEs. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 9-. doi: 10.1186/s41546-016-0007-y [9] Alex Eskin, Maryam Mirzakhani. Counting closed geodesics in moduli space. Journal of Modern Dynamics, 2011, 5 (1) : 71-105. doi: 10.3934/jmd.2011.5.71 [10] Margarida Camarinha, Fátima Silva Leite, Peter Crouch. Riemannian cubics close to geodesics at the boundaries. Journal of Geometric Mechanics, 2022  doi: 10.3934/jgm.2022003 [11] José M. Arrieta, Esperanza Santamaría. Estimates on the distance of inertial manifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 3921-3944. doi: 10.3934/dcds.2014.34.3921 [12] Liliana Trejo-Valencia, Edgardo Ugalde. Projective distance and $g$-measures. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3565-3579. doi: 10.3934/dcdsb.2015.20.3565 [13] Stéphane Sabourau. Growth of quotients of groups acting by isometries on Gromov-hyperbolic spaces. Journal of Modern Dynamics, 2013, 7 (2) : 269-290. doi: 10.3934/jmd.2013.7.269 [14] Penka Georgieva, Aleksey Zinger. Real orientations, real Gromov-Witten theory, and real enumerative geometry. Electronic Research Announcements, 2017, 24: 87-99. doi: 10.3934/era.2017.24.010 [15] Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on Gromov hyperbolic metric graphs. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3725-3757. doi: 10.3934/dcds.2021014 [16] Michael Scheutzow, Maite Wilke-Berenguer. Random Delta-Hausdorff-attractors. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1199-1217. doi: 10.3934/dcdsb.2018148 [17] Marcello Delitala, Tommaso Lorenzi. Evolutionary branching patterns in predator-prey structured populations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2267-2282. doi: 10.3934/dcdsb.2013.18.2267 [18] Alexander Nabutovsky and Regina Rotman. Lengths of geodesics between two points on a Riemannian manifold. Electronic Research Announcements, 2007, 13: 13-20. [19] R. Bartolo, Anna Maria Candela, J.L. Flores. Timelike Geodesics in stationary Lorentzian manifolds with unbounded coefficients. Conference Publications, 2005, 2005 (Special) : 70-76. doi: 10.3934/proc.2005.2005.70 [20] Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75

2020 Impact Factor: 0.929

## Metrics

• PDF downloads (237)
• HTML views (1758)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]