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Canonical coordinates and principal directions for surfaces in \(\mathbb H^2 \times \mathbb R\). (English) Zbl 1241.53010
The authors characterize and classify surfaces in \({\mathbb{H}}^2\times{\mathbb{R}}\) for which \(T\) (the direction of \({\mathbb{R}}\)-fibers) is a principal direction. The case of zero principal curvature corresponds to a constant angle of \(T\) with a unit normal to the surface. The main result reads as: Let \(F : M\to {\mathbb{H}}^2\times{\mathbb{R}}\) be an isometrically immersed surface with \(\theta\neq 0, \frac\pi2\). Then \(M\) has \(T\) as a principal direction if and only if \(F\) is given by \[ F(x; y) =\big(f(y)\cosh \phi(x) + N_f(y)\sinh\phi(x),\;\chi(x)\big), \] where \(f(y)\) is a regular curve in \({\mathbb{H}}^2\) and \(N_f(y)\) represents the normal of \(f\). Moreover, \((\phi, \chi)\) is a regular curve in \({\mathbb{R}}^2\) and the angle function \(\theta\) of this curve is the same as the angle function of the surface parameterized by \(F\).
They study some geometric properties such as minimality and flatness. The study of these surfaces was motivated by the results obtained by F. Dillen, J. Fastenakels and J. Van der Veken [Ann. Global Anal. Geom. 35, No. 4, 381–396 (2009; Zbl 1176.53031)] for the ambient space \({\mathbb{S}}^2\times{\mathbb{R}}\).

53B25 Local submanifolds
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