Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel


CiteScore 2017: 0.65

SCImago Journal Rank (SJR) 2017: 1.063
Source Normalized Impact per Paper (SNIP) 2017: 0.833

Mathematical Citation Quotient (MCQ) 2017: 0.86

Open Access
Online
ISSN
2299-3274
See all formats and pricing
More options …

Geodesics in Asymmetric Metric Spaces

Andrea C. G. Mennucci
Published Online: 2014-05-17 | DOI: https://doi.org/10.2478/agms-2014-0004

Abstract

In a recent paper [17] we studied asymmetric metric spaces; in this context we studied the length of paths, introduced the class of run-continuous paths; and noted that there are different definitions of “length spaces” (also known as “path-metric spaces” or “intrinsic spaces”). In this paper we continue the analysis of asymmetric metric spaces.We propose possible definitions of completeness and (local) compactness.We define the geodesics using as admissible paths the class of run-continuous paths.We define midpoints, convexity, and quasi-midpoints, but without assuming the space be intrinsic.We distinguish all along those results that need a stronger separation hypothesis. Eventually we discuss how the newly developed theory impacts the most important results, such as the existence of geodesics, and the renowned Hopf-Rinow (or Cohn-Vossen) theorem.

Keywords: asymmetric metric; general metric; quasi metric; ostensible metric; Finsler metric; path metric; length space; geodesic curve; Hopf-Rinow theorem

References

  • [1] Luigi Ambrosio and Paolo Tilli. Selected topics in "analysis in metric spaces". Collana degli appunti. Edizioni Scuola Normale Superiore, Pisa, 2000.Google Scholar

  • [2] D. Bao, S. S. Chern, and Z. Shen. An introduction to Riemann-Finsler Geometry. (Springer-Verlag), 2000.Google Scholar

  • [3] Dmitri Burago, Yuri Burago, and Sergei Ivanov. A course in metric geometry, volume 33 of Graduate Studies inMathematics. American Mathematical Society, Providence, RI, 2001.Google Scholar

  • [4] H. Busemann. Local metric geometry. Trans. Amer. Math. Soc., 56:200-274, 1944.CrossrefGoogle Scholar

  • [5] H. Busemann. The geometry of geodesics, volume 6 of Pure and applied mathematics. Academic Press (New York), 1955.Google Scholar

  • [6] H. Busemann. Recent synthetic differential geometry, volume 54 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer Verlag, 1970.Google Scholar

  • [7] S. Cohn-Vossen. Existenz kürzester wege. Compositio math., Groningen, 3:441-452, 1936.Google Scholar

  • [8] J.A Collins and J.B Zimmer. An asymmetric Arzelà-Ascoli theorem. Topology and its Applications, 154(11):2312-2322, 2007.Google Scholar

  • [9] A. Duci and A. Mennucci. Banach-like metrics and metrics of compact sets. 2007.Google Scholar

  • [10] P. Fletcher and W. F. Lindgren. Quasi-uniform spaces, volume 77 of Lecture notes in pure and applied mathematics. Marcel Dekker, 1982.Google Scholar

  • [11] J. L. Flores, J. Herrera, and M. Sánchez. Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds. Mem. Amer. Math. Soc., 226(1064):vi+76, 2013.Google Scholar

  • [12] M. Gromov. Metric Structures for Riemannian and Non-Riemannian Spaces, volume 152 of Progress in Mathematics. Birkhäuser Boston, 2007. Reprint of the 2001 edition.Google Scholar

  • [13] J. C. Kelly. Bitopological spaces. Proc. London Math. Soc., 13(3):71-89, 1963.CrossrefGoogle Scholar

  • [14] H. P. Künzi. Complete quasi-pseudo-metric spaces. Acta Math. Hungar., 59(1-2):121-146, 1992.Google Scholar

  • [15] H. P. Künzi and M. P. Schellekens. On the Yoneda completion of a quasi-metric space. Theoretical Computer Science, 278(1-2):159 - 194, 2002. Mathematical Foundations of Programming Semantics 1996.Google Scholar

  • [16] A. C. G. Mennucci. Regularity and variationality of solutions to Hamilton-Jacobi equations. part ii: variationality, existence, uniqueness. Applied Mathematics and Optimization, 63(2), 2011.Google Scholar

  • [17] A. C. G. Mennucci. On asymmetric distances. Analysis and Geometry in Metric Spaces, 1:200-231, 2013.Google Scholar

  • [18] Athanase Papadopoulos. Metric spaces, convexity and nonpositive curvature, volume 6 of IRMA Lectures in Mathematics and Theoretical Physics. European Mathematical Society (EMS), Zürich, 2005.Google Scholar

  • [19] I. L. Reilly, P. V. Subrahmanyam, and M. K. Vamanamurthy. Cauchy sequences in quasi-pseudo-metric spaces. Monat.Math., 93:127-140, 1982.Google Scholar

  • [20] Riccarda Rossi, Alexander Mielke, and Giuseppe Savaré. A metric approach to a class of doubly nonlinear evolution equations and applications. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7(1):97-169, 2008.Google Scholar

  • [21] M. B. Smyth. Quasi-uniformities: reconciling domains with metric spaces. In Mathematical foundations of programming language semantics (New Orleans, LA, 1987), volume 298 of Lecture Notes in Comput. Sci., pages 236-253. Springer, Berlin, 1988.Google Scholar

  • [22] M. B. Smyth. Completeness of quasi-uniform and syntopological spaces. J. London Math. Soc. (2), 49(2):385-400, 1994.CrossrefGoogle Scholar

  • [23] W. A. Wilson. On quasi-metric spaces. Amer. J. Math., 53(3):675-684, 1931.Google Scholar

  • [24] E. M. Zaustinsky. Spaces with non-symmetric distances. Number 34 in Mem. Amer. Math. Soc. AMS, 1959. Google Scholar

About the article

Received: 2013-10-28

Accepted: 2014-02-26

Published Online: 2014-05-17

Published in Print: 2014-01-01


Citation Information: Analysis and Geometry in Metric Spaces, Volume 2, Issue 1, ISSN (Online) 2299-3274, DOI: https://doi.org/10.2478/agms-2014-0004.

Export Citation

© 2014. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Comments (0)

Please log in or register to comment.
Log in