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Periodica Mathematica Hungarica

, Volume 48, Issue 1–2, pp 69–76 | Cite as

On the Curvature of the Indicatrix Surface in Three-Dimensional Minkowski Spaces

  • Cs. VinczeEmail author
Article

Abstract

As it is well-known a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. If the square of the Minkowski functional is quadratic then we have an Euclidean space and the indicatrix hypersurface S:= F-1 (1) has constant 1 curvature. In his classical paper [1] F. Brickell proved that the converse is also true provided that the indicatrix is symmetric with respect to the origin. M. Ji and Z. Shen investigated the (sectional) curvature of Randers indicatrices and it always turned out greater than zero and less or equal than 1; see [3]. In this note we give a general lower and upper bound for the curvature in terms of the norm of the Cartan tensor.

Mathematics subject classification number

53C60 58B20 

Key words and phrases

Minkowski spaces Cartan tensors curvature 

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References

  1. [1]
    F. Brickell, A theorem on homogeneous functions, J. London Math. Soc. 42 (1967), 325–329.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    C. E. Duran, A volume comparaison theorem for Finsler manifolds, Proc. Amer. Math. Soc., Vol. 126, Number 10, October 1998, 3079–3082.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    M. Ji and Z. Shen, On strongly convex indicatrices in Minkowski geometry, Canad. Math. Bull. 45 (2) (2002), 232–246.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    R. Schneider, Über die Finslerräume mit Sjkl = 0, Arch. Math., Vol. XIX, 1968, 656–658.Google Scholar
  5. [5]
    Z. Shen, On R-quadratic Finsler spaces, Publ. Math. Debrecen 58 (2001), 263–274.MathSciNetzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest 2004

Authors and Affiliations

  1. 1.Institute of Mathematics And InformaticsUniversity Of DebrecenDebrecenHungary

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