A dual rigidity of the sphere and the hyperbolic plane
-
Magdalena Caballero
Abstract
There are several well-known characterizations of the sphere as a regular surface in the Euclidean space. By means of a purely synthetic technique, we get a rigidity result for the sphere without any curvature conditions, completeness or compactness, as well as a dual result for the hyperbolic plane, the spacelike sphere in the Minkowski space.
Funding: The authors are partially supported by Spanish MINECO and ERDF project MTM2013-47828-C2-1-P.
Acknowledgements
The authors thank the referee for his/her helpful and valuable comments.
References
[1] A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. III. Vestnik Leningrad. Univ. 13 (1958), 14–26. MR0102112 Zbl 0269.53024Search in Google Scholar
[2] M. Caballero, R. M. Rubio, Dual characterizations of the sphere and the hyperbolic space in arbitrary dimension. Int. J. Geom. Methods Mod. Phys. 12 (2015), 1560010, 5. MR3400651 Zbl 1332.5300910.1142/S0219887815600105Search in Google Scholar
[3] S.-S. Chern, Some new characterizations of the Euclidean sphere. Duke Math. J. 12 (1945), 279–290. MR0012492 Zbl 0063.0083310.1215/S0012-7094-45-01222-1Search in Google Scholar
[4] J. Hadamard, Sur certaines propriétés des trajectoires en dynamique. Journ. de Math. (5) 3 (1897), 331–387. Zbl 28.0643.01Search in Google Scholar
[5] D. Hilbert, Grundlagen der Geometrie, 3rd ed. Leipzig und Berlin: B. G. Teubner 1909. MR0080308 Zbl 40.0518.01Search in Google Scholar
[6] H. Hopf, Zum Clifford-Kleinschen Raumproblem. Math. Ann. 95 (1926), 313–339. MR1512281 JFM 51.0439.0510.1007/978-3-642-40036-0_1Search in Google Scholar
[7] H. Liebmann, Eine neue Eigenschaft der Kugel. Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. (1899), 44–55. Zbl 30.0557.01Search in Google Scholar
[8] H. Liebmann, Ueber die Verbiegung der geschlossenen Flächen positiver Krümmung. Math. Ann. 53 (1900), 81–112. MR1511083 Zbl 31.0610.0110.1007/BF01456030Search in Google Scholar
[9] R. López, Cyclic surfaces of constant Gauss curvature. Houston J. Math. 27 (2001), 799–805. MR1874672 Zbl 1011.53006Search in Google Scholar
[10] R. López, Surfaces of constant Gauss curvature in Lorentz-Minkowski three-space. Rocky Mountain J. Math. 33 (2003), 971–993. MR2038534 Zbl 1062.5300310.1216/rmjm/1181069938Search in Google Scholar
[11] R. López, Constant mean curvature surfaces with boundary. Springer 2013. MR3098467 Zbl 1278.5300110.1007/978-3-642-39626-7Search in Google Scholar
[12] R. López, Differential geometry of curves and surfaces in Lorentz-Minkowski space. Int. Electron. J. Geom. 7 (2014), 44–107. MR3198740 Zbl 1312.5302210.36890/iejg.594497Search in Google Scholar
[13] W. F. Reynolds, Hyperbolic geometry on a hyperboloid. Amer. Math. Monthly100 (1993), 442–455. MR1215530 Zbl 0789.5102010.1080/00029890.1993.11990430Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- The non-solvable doubly transitive dimensional dual hyperovals
- On the real differential of a slice regular function
- Generalized null 2-type immersions in Euclidean space
- A dual rigidity of the sphere and the hyperbolic plane
- The conjugacy locus of Cayley–Salmon lines
- Steiner’s Porism in finite Miquelian Möbius planes
- Enumeration of complex and real surfaces via tropical geometry
- On complete Yamabe solitons
- Polytopal approximation of elongated convex bodies
- A note on commutative semifield planes
- Quartic surfaces with icosahedral symmetry
Articles in the same Issue
- Frontmatter
- The non-solvable doubly transitive dimensional dual hyperovals
- On the real differential of a slice regular function
- Generalized null 2-type immersions in Euclidean space
- A dual rigidity of the sphere and the hyperbolic plane
- The conjugacy locus of Cayley–Salmon lines
- Steiner’s Porism in finite Miquelian Möbius planes
- Enumeration of complex and real surfaces via tropical geometry
- On complete Yamabe solitons
- Polytopal approximation of elongated convex bodies
- A note on commutative semifield planes
- Quartic surfaces with icosahedral symmetry
