Dupin Hypersurfaces in Lorentzian Space forms

TL;DR

This paper introduces and classifies spacelike Dupin hypersurfaces in Lorentzian space forms using conformal geometry, focusing on those with constant M"{o}bius curvatures, a conformal invariant.

Contribution

It extends the concept of Dupin hypersurfaces to Lorentzian spaces and provides a classification for those with constant M"{o}bius curvatures.

Findings

Classification of spacelike Dupin hypersurfaces with constant M"{o}bius curvatures

Identification of conformal invariants in Lorentzian space forms

Extension of Dupin hypersurface theory to Lorentzian geometry

Abstract

Similar to the definition of Dupin hypersurface in Riemannian space forms, we define the spacelike Dupin hypersurface in Lorentzian space forms. As conformal invariant objects, spacelike Dupin hypersurfaces are studied in this paper using the framework of conformal geometry. Further we classify the spacelike Dupin hypersurfaces with constant M\"{o}bius curvatures, which are the partition ratio of the principal curvatures of the spacelike Dupin hypersurface.