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Surfaces with parallel second fundamental form in Bianchi-Cartan-Vranceanu spaces. (English) Zbl 1029.53071

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, 67-87 (2002).
In this work a complete classification of surfaces with parallel second fundamental form in 3-dimensional Bianchi-Cartan-Vranceanu spaces is given. In two previous works the authors have shown that the only surfaces with parallel second fundamental form in the 3-dimensional Heisenberg group or the special linear group \(\text{SL}_2(\mathbb{R})\) are Hopf cylinders. Noting that both the Heisenberg group and \(\text{SL}_2(\mathbb{R})\) are contact space forms, the authors generalize their previous results by classifying all surfaces with parallel second fundamental form in 3-dimensional Bianchi-Cartan-Vranceanu spaces. Also in this work, explicit matrix group models of 3-dimensional contact spaces forms and explicit models of 3-dimensional homogeneous spaces with Bianchi-Cartan-Vranceanu metric and their associated almost contact structures are exhibited.
For the entire collection see [Zbl 1007.00038].

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53D15 Almost contact and almost symplectic manifolds
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