Mixed type surfaces with bounded Gaussian curvature in three-dimensional Lorentzian manifolds

2018/11/28 by Atsufumi Honda, Kentaro Saji, Honda, Atsufumi +3 · 2 citations
Mathematics · #35M10 #53A35 #Differential Geometry (math.DG) #FOS: Mathematics #Primary 53B30 #Secondary 57R45 #math.DG #msc:35M10 #msc:53A35 #msc:53B30 #msc:57R45

paper · pdf · doi:10.48550/arxiv.1811.11392

34 pages, 3 figures

arxiv created 2019/11/23 · arxiv updated 2019/11/26

Abstract

A mixed type surface is a connected regular surface in a Lorentzian 3-manifold with non-empty spacelike and timelike point sets. The induced metric of a mixed type surface is a signature-changing metric, and their lightlike points may be regarded as singular points of such metrics. In this paper, we investigate the behavior of Gaussian curvature at a non-degenerate lightlike point of a mixed type surface. To characterize the boundedness of Gaussian curvature at a non-degenerate lightlike points, we introduce several fundamental invariants along non-degenerate lightlike points, such as the lightlike singular curvature and the lightlike normal curvature. Moreover, using the results by Pelletier and Steller, we obtain the Gauss-Bonnet type formula for mixed type surfaces with bounded Gaussian curvature.

Cited by

• Envelopes created by curve families in the Lorentz-Minkowski plane
• Behavior of Gauss curvatures and mean curvatures of Lightcone framed surfaces in the Lorentz-Minkowski 3-space

Related

• Isometric deformations of mixed type surfaces in Lorentz-Minkowski space
• Mixed type surfaces with bounded mean curvature in 3-dimensional space-times
• Behavior of Gauss curvatures and mean curvatures of Lightcone framed surfaces in the Lorentz-Minkowski 3-space
• Behavior of the Gaussian curvature of timelike minimal surfaces with singularities
• Geometry of lightlike locus on mixed type surfaces in Lorentz-Minkowski 3-space from a contact viewpoint