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Differential geometry of immersed surfaces in three-dimensional normed spaces

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Abstract

In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff orthogonality, we get an analogue of the Gauss map. Then we can define concepts of principal, Gaussian, and mean curvatures in terms of the eigenvalues of the differential of this map. Considering planar sections containing the normal field, we also define normal curvatures at each point of the surface, and with respect to each tangent direction. We investigate the relations between these curvature types. Further on we prove that, under an additional hypothesis, a compact, connected surface without boundary whose Minkowski Gaussian curvature is constant must be a Minkowski sphere. Since existing literature on the subject of our paper is widely scattered, in the introductory part also a survey of related results is given.

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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. Antonelli, P.L.: Handbook of Finsler Geometry. Kluwer Academic Publishers, Boston (2003)

    Book  MATH  Google Scholar 

  3. Balestro, V., Martini, H., Shonoda, E.: Concepts of curvatures in normed planes. Expo. Math. 37(4), 347–281 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Balestro, V., Martini, H., Teixeira, R.: Surface immersions in normed spaces from the affine point of view. Geom. Dedicata 201(1), 21–31 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Balestro, V., Martini, H., Teixeira, R.: On curvature of immersed surfaces in normed spaces. Monatsh. Math. 192(2), 291–309 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  6. Balestro, V., Martini, H., Teixeira, R.: Some topics in differential geometry of normed spaces. Adv. Geom. (2020, to appear), arXiv: 1709.01399

  7. Barthel, W., Kern, U.: Affine und relative Differentialgeometrie. In: O. Giering and J. Hoschek. Geometrie und ihre Anwendungen. Hanser, München, pp 283–317, (1994)

    MATH  Google Scholar 

  8. Biberstein O. A.: Elements de Géométrie Différentielle Minkowskienne. PhD thesis, Université de Montreal (1957)

  9. Blaschke, W.: Gesammelte Werke, Band 4: Affine Differentialgeometrie. Differentialgeometrie der Kreis- und Kugelgruppen. Collected works, Vol. 4 (in German). In: Burau, W., Chern, S.S., Leichtweiss, K., Müller, H.R., Santaló, L.A., Simon, U., Strubecker, K. (Eds.) With commentaries by W. Burau and U. Simon. Thales–Verlag, Essen (1985)

  10. Borisenko, A.A., Tenenblat, K.: On the total curvature of curves in a Minkowski space. Israel J. Math. 191(2), 755–769 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Cartan, E.: Sur les espaces de Finsler. C. R. Acad. Sci. Paris 196, 582–586 (1933)

    MATH  Google Scholar 

  14. Duschek, A.: Über relative Flächentheorie. Sitzungsber. Akad. Wiss. Wien, 135:1–8, Abt. Ila (1926)

  15. Finsler, P.: Über Kurven und Flächen in allgemeinen Räumen. Dissertation, Göttingen, 1918 (reprinted by Birkhäuser) (1951)

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

    Article  MathSciNet  MATH  Google Scholar 

  17. He, Q., Yin, S., Shen, Y.: Isoparametric hypersurfaces in Minkowski spaces. Differ. Geom. Appl. 47, 133–158 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Horváth, A.G.: Semi-indefinite inner product and generalized Minkowski spaces. J. Geom. Phys. 60, 1190–1208 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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 

  20. 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 

  21. Minkowski, H.: Geometrie der Zahlen. Teubner, Leipzig (1896)

    MATH  Google Scholar 

  22. Müller, E.: Relative Minimalflächen. Monatsh. Math. 31, 3–19 (1921)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  24. Perdigão do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Upper Saddle River (1976)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  26. Riemann, B.: Über die Hypothesen, welche der Geometrie zu Grunde liegen. Abh. Königl. Ges. Wiss. Göttingen 13, 133–152 (1868)

  27. Rund, H.: Zur Begründung der Differentialgeometrie der Minkowskischen Räume. Arch. Math. 3, 60–69 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  28. Schneider, R.: Differentialgeometrie im Grossen. I. Math. Z. 101, 375–406 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shirokov, P.A., Shirokov, A.P.: Affine Differentialgeometrie. B. G. Teubner, Leipzig (1962)

    MATH  Google Scholar 

  30. Simon, U., Schwenk-Schellschmidt, A., Viesel, H.: Introduction to the Affine Differential Geometry of Hypersurfaces. Science University of Tokyo, Tokyo (1991)

    MATH  Google Scholar 

  31. Süss, W.: Zur relativen Differentialgeometrie I: Über Eilinien und Eiflächen in der elementaren und affinen Differentialgeometrie. Jpn. J. Math. 4, 57–75 (1927)

    Article  MATH  Google Scholar 

  32. Süss, W.: Zur relativen Differentialgeometrie V: Über Eihyperflächen im \({\mathbb{R}}^{n+1}\). Tôhoku J. Math. 31, 202–209 (1927)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  34. Witt, R.: Eine relativgeometrische Erweiterung der affinen Flächentheorie. Compos. Math. 1, 429–447 (1935)

    MATH  Google Scholar 

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Acknowledgements

The authors wish to thank one referee for drawing their attention to the paper [17].

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Correspondence to Vitor Balestro.

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Communicated by Vicente Cortés.

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Balestro, V., Martini, H. & Teixeira, R. Differential geometry of immersed surfaces in three-dimensional normed spaces. Abh. Math. Semin. Univ. Hambg. 90, 111–134 (2020). https://doi.org/10.1007/s12188-020-00219-7

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  • DOI: https://doi.org/10.1007/s12188-020-00219-7

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