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Surfaces with prescribed Weingarten operator. (English) Zbl 1029.53005

Opozda, Barbara (ed.) et al., PDEs, submanifolds and affine differential geometry. Contributions of a conference, Warsaw, Poland, September 4-10, 2000. Warsaw: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 57, 171-178 (2002).
The authors study surfaces \(x,x^\#:M\to E^3\) with the property that \(x\) is a surface of revolution and that the respective Weingarten operators \(S,S^\#\) coincide. It has been shown before that \(x,x^\#\) are congruent if (i) \(x,x^\#\) are ovaloids, and (ii) \(x\) is a surface of revolution with nowhere dense umbilics. Here the authors extend this result to the case of compact \(M\) with genus zero, if some technical condition is met. The method of proof is a discussion of several local cases, which shows that \(x\) is also a surface of revolution, and which yields examples of families of non-isometric surfaces with the same Weingarten operator.
For the entire collection see [Zbl 1007.00038].

MSC:

53A05 Surfaces in Euclidean and related spaces
53C24 Rigidity results
53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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