On Weingarten surfaces in Euclidean and Lorentzian 3-space

Guilfoyle, Brendan; Klingenberg, Wilhelm

Abstract

We study the neutral Kähler metric on the space of time-like lines in Lorentzian View the MathML source, which we identify with the total space of the tangent bundle to the hyperbolic plane. We find all of the infinitesimal isometries of this metric, as well as the geodesics, and interpret them in terms of the Lorentzian metric on View the MathML source. In addition, we give a new characterisation of Weingarten surfaces in Euclidean E3 and Lorentzian View the MathML source as the vanishing of the scalar curvature of the associated normal congruence in the space of oriented lines. Finally, we relate our construction to the classical Weierstrass representation of minimal and maximal surfaces in E3 and View the MathML source.

Citation

Guilfoyle, B., & Klingenberg, W. (2010). On Weingarten surfaces in Euclidean and Lorentzian 3-space. Differential Geometry and its Applications, 28(4), 454-468. https://doi.org/10.1016/j.difgeo.2009.12.002

• Journal Article Type: Article
• Publication Date: 2010-08
• Deposit Date: Mar 7, 2011
• Journal: Differential Geometry and its Applications
• Print ISSN: 0926-2245
• Electronic ISSN: 1872-6984
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 28
• Issue: 4
• Pages: 454-468
• DOI: https://doi.org/10.1016/j.difgeo.2009.12.002
• Public URL: https://durham-repository.worktribe.com/output/1534056