Explicit examples to the H-problem of Heinz Hopf. (English) Zbl 0615.53050

The author determines explicitly (by Jacobi elliptic functions) all immersions of a torus into \({\mathbb{R}}^ 3\) with one family of curvature lines plane and the other (then necessarily) spherical. There are a countable set of nonisomorphic real analytic immersions. The paper is a beautiful example of practical computation in threedimensional differential geometry, with well drawn figures. It should be pointed out that all surfaces abtained by the author have constant mean curvature. The tori so defined are the first known surfaces of constant mean curvature and genus \(>0\).
Reviewer: H.Guggenheimer


53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A05 Surfaces in Euclidean and related spaces
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