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On the Gauss map of ruled surfaces

Published online by Cambridge University Press:  18 May 2009

Christos Baikoussis  and
David E. Blair
Christos Baikoussis
Affiliation:
Department of Mathematics, University of Ioannina, Ioannina 45110, Greece
David E. Blair
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.A.
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Let M2 be a (connected) surface in Euclidean 3-space E3, and let G:M2→S2(1) ⊂ E3 be its Gauss map. Then, according to a theorem of E. A. Ruh and J. Vilms [3], M2 is a surface of constant mean curvature if and only if, as a map from M2 to S2(1), G is harmonic, or equivalently, if and only if

where δ is the Laplace operator on M2 corresponding to the induced metric on M2 from E3 and where G is seen as a map from M2to E3. A special case of (1.1) is given by

i.e., the case where the Gauss map G:M2→E3 is an eigenfunction of the Laplacian δ on M2.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 34 , Issue 3 , September 1992 , pp. 355 - 359
Copyright
Copyright © Glasgow Mathematical Journal Trust 1992

References

REFERENCES

1.Chen, B. Y., Dillen, F., Verstraelen, L. and Vrancken, L., Ruled surfaces of finite type, Bull. Austral. Math. Soc. 42 (1990), 447–453.CrossRefGoogle Scholar
2.Dillen, F., Pas, J. and Verstraelen, L., On the Gauss map of surfaces of revolution, Bull. Inst. Math. Acad. Sinica, 18 (1990), 239–246.Google Scholar
3.Ruh, E. A. and Vilms, J., The tension field of the Gauss map, Trans. Amer. Math. Soc. 149 (1970), 569–573.CrossRefGoogle Scholar