Georgiou, Nikos; Guilfoyle, Brendan A characterization of Weingarten surfaces in hyperbolic 3-space. (English) Zbl 1208.53010 Abh. Math. Semin. Univ. Hamb. 80, No. 2, 233-253 (2010). The study of Weingarten surfaces is a classical topic in differential geometry, as introduced by Weingarten in 1861. A surface \(S\) in a 3-dimensional Euclidean space \(\mathbb R^3\) is called a Weingarten surface if there is some relation between its two principal curvatures \(k_1\) and \(k_2\), that is, if there is a smooth function \(F\) of two variables such that \(F(k_1,k_2)=0\). In particular, if \(K\) and \(H\) denote the Gauss and the mean curvature of \(S\), respectively, the identity \(F(k_1,k_2)=0\) implies a relation \(\Gamma(K,H)=0\). We need to know some examples of surfaces and relations between their curvatures as models for classifications.In this paper, the authors give a new characterization of the Weingarten condition for surfaces in \(\mathbb H^3\), which means to classify the totally null surfaces in L(\(\mathbb H^3\)) and recover the well known holomorphic constructions of flat and CMC 1 surfaces in \(\mathbb H^3\). Moreover, in this paper, they study 2-dimensional submanifolds of the space L(\(\mathbb H^3\)) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral structure. Then, the authors prove that the induced metric on a Lagrangian surface in L(\(\mathbb H^3\)) has zero Gauss curvature iff the orthogonal surfaces in \(\mathbb H^3\) are Weingarten. Reviewer: Rüstem Kaya (Eskişehir) Cited in 7 Documents MSC: 53A35 Non-Euclidean differential geometry 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) Keywords:Kähler structure; hyperbolic 3-space; Weingarten surfaces PDFBibTeX XMLCite \textit{N. Georgiou} and \textit{B. Guilfoyle}, Abh. Math. Semin. Univ. Hamb. 80, No. 2, 233--253 (2010; Zbl 1208.53010) Full Text: DOI arXiv References: [1] Bryant, R.L.: Surfaces of mean curvature one in hyperbolic space. Astérisque 154(5), 321–347 (1987) [2] Chern, S.: Some new characterizations of the Euclidean sphere. Duke Math. J. 12, 279–290 (1945) · Zbl 0063.00833 [3] Epstein, C.L.: Envelopes of horospheres and Weingarten surfaces in hyperbolic 3-space. Preprint [4] Georgiou, N., Guilfoyle, B.: On the space of oriented geodesics of hyperbolic 3-space. Rocky M. J. Math. 40, 1183–1219 (2010) · Zbl 1202.53045 [5] Georgiou, N., Guilfoyle, B., Klingenberg, W.: Totally null surfaces in neutral Kähler 4-manifolds. arXiv:0810.4054 [math.DG] (2008) · Zbl 1162.53016 [6] Hartman, P., Witner, A.: Umbilical points and W-surfaces. Am. J. Math. 76, 502–508 (1954) · Zbl 0055.39601 [7] Kokubu, M., Umehara, M., Yamada, K.: Flat fronts in hyperbolic 3-space. Pac. J. Math. 216, 149–176 (2004) · Zbl 1078.53009 [8] Kokubu, M., Rossman, W., Saji, K., Umehara, M., Yamada, K.: Singularities of flat fronts in hyperbolic 3-space. Pac. J. Math. 221, 303–351 (2005) · Zbl 1110.53044 [9] Kühnel, W., Steller, M.: On closed Weingarten surfaces. Monatshefte. Math. 146, 113–126 (2005) · Zbl 1093.53004 [10] Penrose, R., Rindler, W.: Spinors and Spacetime, vol. 2. Cambridge University Press, Cambridge (1986) · Zbl 0591.53002 [11] Roitman, P.: Flat surfaces in hyperbolic space as normal surfaces to a congruence of geodesics. Tohoku Math. J. 59, 21–37 (2007) · Zbl 1140.53004 [12] Salvai, M.: On the geometry of the space of oriented lines of hyperbolic space. Glasg. Math. J. 49, 357–366 (2007) · Zbl 1130.53013 [13] Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. IV. Publish or Perish, Houston (1999) · Zbl 1213.53001 [14] Weingarten, J.: Ueber eine Klasse auf einander abwickelbarer Flächen. J. Reine Angew. Math. 59, 382–393 (1861) · ERAM 059.1573cj This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.