Spheres and Tori as Elliptic Linear Weingarten Surfaces

Abstract

The linear Weingarten condition with ellipticity for the mean curvature and the extrinsic Gaussian curvature on a surface in the three-sphere can define a Riemannian metric which is called the elliptic linear Weingarten metric. We established some local characterizations of the round spheres and the tori immersed in the 3-dimensional unit sphere, along with the Laplace operator, the spherical Gauss map and the Gauss map associated with the elliptic linear Weingarten metric.

Keywords

• elliptic linear Weingarten metric
• finite-type immersion
• spherical Gauss map
• isoparametric surface
• torus