Extrinsic sphere and totally umbilical submanifolds in Finsler spaces

Document Type: Research Paper

Authors

Faculty of‎ ‎Mathematics and Computer Science‎, ‎Amirkabir University of Technology (Tehran Polytechnic)‎, ‎15914‎, ‎Tehran‎, ‎Iran.

Abstract

‎Based on a definition for circle in Finsler space‎, ‎recently proposed by one of the present authors and Z‎. ‎Shen‎, ‎a natural definition of extrinsic sphere in Finsler geometry is given and it is shown that a connected submanifold of a Finsler manifold is totally umbilical and has non-zero parallel mean curvature vector field‎, ‎if and only if its circles coincide with circles of the ambient manifold‎.
Finally‎, ‎some examples of extrinsic sphere in Finsler geometry‎, ‎particularly in Randers spaces are given.

Keywords

Main Subjects


L.M‎. ‎Abatangelo‎, ‎On totally umbilical submanifolds of locally Minkowski manifold‎, ‎ Collect‎. ‎Math. 43 (1992)‎, ‎no‎. ‎2‎, ‎151--175‎.

H‎. ‎Akbar-Zadeh‎, ‎Initiation to Global Finslerian Geometry‎, ‎Vol‎. ‎68‎, ‎Elsevier Science‎, ‎2006‎.

D‎. ‎Bao‎, ‎S‎. ‎Chern and Z‎. ‎Shen‎, ‎An introduction to Riemann-Finsler Geometry‎. ‎Grad‎. ‎Texts in Math‎. ‎200‎, ‎Springer‎, ‎2000‎.

‎A‎. ‎Bejanco and H.R‎. ‎Farran‎, ‎Geometry of pseudo-Finsler submanifolds‎, ‎Kluwer Acad‎. ‎Publ.‎, ‎Dordrecht‎, ‎2000‎.

‎B‎. ‎Bidabad‎, ‎A classification of complete Finsler manifolds through the conformal theory of curves‎, ‎ Differential Geom‎. ‎Appl. 35 (2014) 350--360‎.

B‎. ‎Bidabad and Z‎. ‎Shen‎, ‎Circle-preserving transformations in Finsler spaces‎, ‎ Publ‎. ‎Math‎. ‎Debrecen, 81 (2012)‎, ‎no‎. ‎3-4‎, ‎435--445‎.

‎I‎. ‎Bucataru and R‎. ‎Miron‎, ‎Finsler-Lagrange Geometry‎, ‎Applications to Dynamical Systems‎, ‎Editura Academiei Române‎, ‎Bucuresti‎, ‎2007‎.

‎Q‎. ‎He‎, ‎W‎. ‎Yang and W‎. ‎Zhao‎, ‎On totally umbilical submanifolds of Finsler spaces‎, ‎ Ann‎. ‎Polon‎. ‎Math. 100 (2011)‎, ‎no‎. ‎2‎, ‎147--157‎.

S‎. ‎Kobayashi and K‎. ‎Nomizu‎, ‎Foundations of Differential Geometry‎, ‎Vol‎. ‎I‎, ‎Vol‎. ‎II‎, ‎Wiley-Interscience‎, ‎New York‎, ‎1963‎, ‎1969‎.

J‎. ‎Li‎, ‎Umbilical hypersurfaces of Minkowski spaces‎, ‎ Math‎. ‎Commun. 17 (2012)‎, ‎no‎. ‎1‎, ‎63--70‎.

K‎. ‎Nomizu‎, ‎Generalized central spheres and the notion of spheres in Riemannian geometry‎, ‎ Tohoku Math‎. ‎J. 25 (1973) 129--137‎.

K‎. ‎Nomizu and K‎. ‎Yano‎, ‎On circles and spheres in Riemannian geometry‎, ‎ Math‎. ‎Ann. 210 (1974)‎, ‎no‎. ‎2‎, ‎163--170‎.

‎M.K‎. ‎Sedaghat and B‎. ‎Bidabad‎, ‎The Axiom of Spheres in Finsler Geometry‎, Annals Un‎. ‎Al‎. ‎I‎. ‎Cuza Iasi Mat. 3 (2016)‎, ‎no‎. ‎8‎, ‎883--890‎.

M.K‎. ‎Sedaghat and B‎. ‎Bidabad‎, ‎On a class of complete Finsler manifolds‎, ‎ Internat‎. ‎J‎. ‎Math. 26 (2015)‎, ‎no‎. ‎10‎, ‎Article ID 1550091‎, ‎9 pages‎.


Volume 43, Issue 2
March and April 2017
Pages 337-347
  • Receive Date: 18 January 2015
  • Revise Date: 17 November 2015
  • Accept Date: 17 November 2015