On quasicomplete \(k\)-surfaces in 3-dimensional space-forms. (English) Zbl 1536.53017
Let \(e:S\rightarrow X\) be an immersed surface in some 3-dimensional space-form \(X_{c}\) of constant sectional curvature \(c\), and let \(\mathrm{I}_{e}\) and \(\mathrm{III}_{e}\) denote its first and third fundamental form. The immersed surface \((S,e)\) is said to be quasicomplete whenever the Riemannian metric \(\mathrm{I}_{e}+\mathrm{III}_{e}\) is complete.
The main result of the paper extends a classical theorem of Liebmann (see [H. Hopf, Differential geometry in the large. Seminar lectures New York University 1946 and Stanford University 1956. With a preface by S. S. Chern. 2nd ed. Berlin etc.: Springer-Verlag (1989; Zbl 0669.53001)]) to quasi-complete surfaces:
Theorem. If \(k>\max(0,-c)\), then the only quasicomplete surfaces in \(X_{c}\) with constant extrinsic curvature \(k\) are geodesic spheres.
The main result of the paper extends a classical theorem of Liebmann (see [H. Hopf, Differential geometry in the large. Seminar lectures New York University 1946 and Stanford University 1956. With a preface by S. S. Chern. 2nd ed. Berlin etc.: Springer-Verlag (1989; Zbl 0669.53001)]) to quasi-complete surfaces:
Theorem. If \(k>\max(0,-c)\), then the only quasicomplete surfaces in \(X_{c}\) with constant extrinsic curvature \(k\) are geodesic spheres.
Reviewer: Andrea Tamburelli (Houston)
MSC:
| 53A05 | Surfaces in Euclidean and related spaces |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
Citations:
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