Skip to main content
SpringerLink
Log in

Surface immersions in normed spaces from the affine point of view

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

The aim of this paper is to investigate the differential geometry of immersed surfaces in three-dimensional normed spaces from the viewpoint of affine differential geometry. We endow the surface with a useful Riemannian metric which is closely related to normal curvature, and from this we re-calculate the Minkowski Gaussian and mean curvatures. These curvatures are also re-obtained in terms of ambient affine distance functions, and as a consequence we characterize minimal surfaces as the solutions of a certain differential equation. We also investigate in which cases it is possible that the affine normal and the Birkhoff normal vector fields of an immersion coincide, proving that this only happens when the geometry is Euclidean.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alonso, J., Martini, H., Wu, S.: On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces. Aequ. Math. 83(1–2), 153–189 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balestro, V., Martini, H., Shonoda, E.: Concepts of curvatures in normed planes. Expo. Math. arXiv:1702.01449 (2017)

  3. Balestro, V., Martini, H., Teixeira, R.: Differential geometry of immersed surfaces in three-dimensional normed spaces. Preprint, submitted. arXiv:1707.04226 (2017)

  4. Balestro, V., Martini, H., Teixeira, R.: Some topics in differential geometry of normed spaces. Preprint, submitted. arXiv:1709.01399 (2017)

  5. Blaschke, W.: Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. II. Affine Differentialgeometrie. Grundlehren der mathematischen Wissenschaften in Einzeldarstellung mit besonderer Berücksichtigung der Anwendungsgebiete, Band VII. Springer, Berlin (1923)

  6. Busemann, H.: The foundations of Minkowskian geometry. Comment. Math. Helvet. 24(1), 156–187 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  7. Busemann, H.: The geometry of Finsler spaces. Bull. Am. Math. Soc. 56(1), 5–16 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  8. Busemann, H.: On normal coordinates in Finsler spaces. Math. Ann. 129, 417–423 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  9. Davis, D.: Affine normal curvature of hypersurfaces from the point of view of singularity theory. Geom. Dedicata 141, 137–145 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Deicke, A.: Über die Finsler–Räume mit \({A}_i = 0\). Arch. Math. 4, 45–51 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guggenheimer, H.: Pseudo-Minkowski differential geometry. Ann. Mat. Pura Appl. (4) 70(1), 305–370 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  12. Loftin, J.: Survey on affine spheres. In: Ji, L. et al. (eds.) Handbook of Geometric Analysis, No. 2. Advanced Lectures in Mathematics, vol. 13, pp. 161–192. International Press, Boston (2010)

  13. Martini, H., Swanepoel, K.J.: The geometry of Minkowski spaces—a survey. Part II. Expo. Math. 22(2), 93–144 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Martini, H., Swanepoel, K.J., Weiß, G.: The geometry of Minkowski spaces—a survey. Part I. Expo. Math. 19(2), 97–142 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nomizu, K., Sasaki, T.: Affine Differential Geometry. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  16. do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood-Cliffs, NJ (1976)

    MATH  Google Scholar 

  17. Petty, C.M.: On the geometry of the Minkowski plane. Riv. Mat. Univ. Parma 6, 269–292 (1955)

    MathSciNet  MATH  Google Scholar 

  18. Shen, Y.-B., Shen, Z.: Introduction to Modern Finsler Geometry. World Scientific Publishing Co., Singapore (2016)

    Book  MATH  Google Scholar 

  19. Shen, Z.: Lectures on Finsler Geometry. World Scientific Publishing Co., Singapore (2001)

    Book  MATH  Google Scholar 

  20. Thompson, A.C.: Minkowski Geometry. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CEFET/RJ Campus Nova Friburgo, Nova Friburgo, 28635000, Brazil

    Vitor Balestro

  2. Fakultät für Mathematik, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    Horst Martini

  3. Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, 24210201, Brazil

    Ralph Teixeira

Authors
  1. Vitor Balestro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Horst Martini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ralph Teixeira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vitor Balestro.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Balestro, V., Martini, H. & Teixeira, R. Surface immersions in normed spaces from the affine point of view. Geom Dedicata 201, 21–31 (2019). https://doi.org/10.1007/s10711-018-0380-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-018-0380-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation