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Tytuł artykułu

Geodesics in Asymmetic Metric Spaces

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Rocznik
Tom
2
Numer
1
Opis fizyczny
Daty
wydano
2014-01-01
otrzymano
2013-10-28
zaakceptowano
2014-02-26
online
2014-05-17
Twórcy
Bibliografia
  • [1] Luigi Ambrosio and Paolo Tilli. Selected topics in "analysis in metric spaces". Collana degli appunti. Edizioni Scuola Normale Superiore, Pisa, 2000.
  • [2] D. Bao, S. S. Chern, and Z. Shen. An introduction to Riemann-Finsler Geometry. (Springer-Verlag), 2000.
  • [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.
  • [4] H. Busemann. Local metric geometry. Trans. Amer. Math. Soc., 56:200-274, 1944.
  • [5] H. Busemann. The geometry of geodesics, volume 6 of Pure and applied mathematics. Academic Press (New York), 1955.
  • [6] H. Busemann. Recent synthetic differential geometry, volume 54 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer Verlag, 1970.
  • [7] S. Cohn-Vossen. Existenz kürzester wege. Compositio math., Groningen, 3:441-452, 1936.
  • [8] J.A Collins and J.B Zimmer. An asymmetric Arzelà-Ascoli theorem. Topology and its Applications, 154(11):2312-2322, 2007.
  • [9] A. Duci and A. Mennucci. Banach-like metrics and metrics of compact sets. 2007.
  • [10] P. Fletcher and W. F. Lindgren. Quasi-uniform spaces, volume 77 of Lecture notes in pure and applied mathematics. Marcel Dekker, 1982.
  • [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.
  • [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.
  • [13] J. C. Kelly. Bitopological spaces. Proc. London Math. Soc., 13(3):71-89, 1963.
  • [14] H. P. Künzi. Complete quasi-pseudo-metric spaces. Acta Math. Hungar., 59(1-2):121-146, 1992.
  • [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.
  • [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.[WoS]
  • [17] A. C. G. Mennucci. On asymmetric distances. Analysis and Geometry in Metric Spaces, 1:200-231, 2013.
  • [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.
  • [19] I. L. Reilly, P. V. Subrahmanyam, and M. K. Vamanamurthy. Cauchy sequences in quasi-pseudo-metric spaces. Monat.Math., 93:127-140, 1982.
  • [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.
  • [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.
  • [22] M. B. Smyth. Completeness of quasi-uniform and syntopological spaces. J. London Math. Soc. (2), 49(2):385-400, 1994.[Crossref]
  • [23] W. A. Wilson. On quasi-metric spaces. Amer. J. Math., 53(3):675-684, 1931.[Crossref]
  • [24] E. M. Zaustinsky. Spaces with non-symmetric distances. Number 34 in Mem. Amer. Math. Soc. AMS, 1959.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_agms-2014-0004
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