Abstract
A representation of the Joachimsthal surfaces (having a family of curvature lines that lie in totally geodesic 2-spheres) in the sphereS 3 is obtained. It is proved that, if a surface of constant mean curvature inS 3 has one family of curvature lines lying in totally geodesic 2-spheres, then it is a surface of rotation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Bianchi,Lezioni di geometria differenziale. 1, Edit. Nic. Zanichelli, Bologna (1927).
V. I. Shulikovskii,The Classical Differential Geometry [in Russian], Fizmatgiz, Moscow (1963).
H. C. Wente, “Counterexample to a conjecture of H. Hopf,”Pacific J. Math.,121, No. 1, 193–243 (1986).
U. Abresh, “Constant mean curvature tori in terms of elliptic functions,”J. Reine Angew. Math.,374, 169–172 (1987).
R. Walter, “Explicit examples to the H-problem of Heinz Hopf,”Geom. Dedicata,23, 187–213 (1987).
A. I. Bobenko, “All constant mean curvature tori in ℝ3,S 3,H 3 in terms of theta-functions,”Math. Ann.,290, 209–245 (1991).
A. I. Bobenko, “Surfaces of constant mean curvature and integrable equations,”Uspekhi Mat. Nauk [Russian Math. Surveys],46, No. 4, 3–42 (1991).
H. C. Wente, “Constant mean curvature immersions of Enneper type,”Memoirs. Amer. Math. Soc., No. 478 (1992).
H. C. Wente, “Complete immersions of constant mean curvature,”Proc. Symp. Pure Math.,54, No. 1, 497–512 (1993).
Author information
Authors and Affiliations
Additional information
Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 221–229, February, 2000.
Rights and permissions
About this article
Cite this article
Masal’tsev, L.A. Joachimsthal surfaces inS 3 . Math Notes 67, 176–182 (2000). https://doi.org/10.1007/BF02686244
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02686244