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Abstract
In this paper, we introduce the fourth fundamental forms for hypersurfaces in Hn+1 and space-like hypersurfaces in S1n+1, and discuss the conformality of the normal Gauss map of the hypersurfaces in Hn+1 and S1n+1. Particularly, we discuss the surfaces with conformal normal Gauss map in H³ and S³1, and prove a duality property. We give a Weierstrass representation formula for space-like surfaces in S³1 with conformal normal Gauss map. We also state the similar results for time-like surfaces in S³1. Some examples of surfaces in S³1 with conformal normal Gauss map are given and a fully nonlinear equation of Monge-Ampère type for the graphs in S³1 with conformal normal Gauss map is derived.
Highlights
It is well known that the classical Gauss map has played an important role in the study of the surface theory in R3 and has been generalized to the submanifold of arbitrary dimension and codimension immersed into the space forms with constant sectional curvature.for the n-dimensional submanifold x : M → V in space V with constant sectional curvature, Obata (Obata 1968) introduced the generalized Gauss map which assigns each point p of M to the totally geodesic n-subspace of V tangent to x(M) at x( p)
Using the Weierstrass representation formula, Bryant studied the properties of constant mean curvature one surfaces
Gálvez and Martínez (Gálvez and Martínez 2000) studied the properties of the Gauss map of a surface immersed into the Euclidean 3-space R3 by using the second conformal structure on surface, and obtained a Weierstrass representation formula for surfaces with prescribed Gauss map
Summary
It is well known that the classical Gauss map has played an important role in the study of the surface theory in R3 and has been generalized to the submanifold of arbitrary dimension and codimension immersed into the space forms with constant sectional curvature (see Osserman 1980). Kokubu (Kokubu 1997) considered the n-dimensional hyperbolic space H n as a Lie group G with a left-invariant metric, and defined the normal Gauss map of the surfaces which assigns each point of the surface to the tangent plane left translated to the Lie algebra of G He gave a Weierstrass representation formula for minimal surfaces in H n. Gálvez and Martínez (Gálvez and Martínez 2000) studied the properties of the Gauss map of a surface immersed into the Euclidean 3-space R3 by using the second conformal structure on surface, and obtained a Weierstrass representation formula for surfaces with prescribed Gauss map. We state the similar results for time-like surfaces in S13 with conformal normal Gauss map
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