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Isothermic surfaces and the Gauss map. (English) Zbl 0692.53003

Summary: We give a necessary and sufficient condition for the Gauss map of an immersed surface M in n-space to arise simultaneously as the Gauss map of an anti-conformal immersion of M into the same space. The condition requires that the lines of curvature of each normal section lie on the zero set of a harmonic function. The result is applied to a class of surfaces studied by S. S. Chern which admit an isometric deformation preserving the principal curvatures.

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53C40 Global submanifolds
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References:

[1] Shiing Shen Chern, Deformation of surfaces preserving principal curvatures, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 155 – 163. · Zbl 0566.53002
[2] Luther Pfahler Eisenhart, A treatise on the differential geometry of curves and surfaces, Dover Publications, Inc., New York, 1960. · Zbl 0090.37803
[3] David A. Hoffman and Robert Osserman, The Gauss map of surfaces in \?\(^{n}\), J. Differential Geom. 18 (1983), no. 4, 733 – 754 (1984). · Zbl 0535.53004
[4] David A. Hoffman and Robert Osserman, The Gauss map of surfaces in \?³ and \?\(^{4}\), Proc. London Math. Soc. (3) 50 (1985), no. 1, 27 – 56. · Zbl 0535.53005 · doi:10.1112/plms/s3-50.1.27
[5] Heinz Hopf, Differential geometry in the large, Lecture Notes in Mathematics, vol. 1000, Springer-Verlag, Berlin, 1983. Notes taken by Peter Lax and John Gray; With a preface by S. S. Chern. · Zbl 0526.53002
[6] Katsuei Kenmotsu, The Weierstrass formula for surfaces of prescribed mean curvature, Minimal submanifolds and geodesics (Proc. Japan-United States Sem., Tokyo, 1977) North-Holland, Amsterdam-New York, 1979, pp. 73 – 76. Katsuei Kenmotsu, Weierstrass formula for surfaces of prescribed mean curvature, Math. Ann. 245 (1979), no. 2, 89 – 99. · Zbl 0402.53002 · doi:10.1007/BF01428799
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