Surfaces in a pseudo-sphere with harmonic or 1-type pseudo-spherical Gauss map

Abstract

Abstract. We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification is obtained, while in other cases the solutions are described by an explicit system of partial differential equations.

Differential Geometry of 1-type Submanifolds and Submanifolds with 1-type Gauss Map

“…Theorem 14.6. [15] Let M be a non-harmonic space-like surface in S m s (1). Then M has 1-type pseudo-spherical Gauss map if and only if it is an open part of the Clifford torus lying fully in a totally geodesic 3-sphere S 3 (1) ⊂ S m s (1).…”

“…The authors of [15] also classified harmonic space-like and harmonic Lorenztian surfaces in S m s (1) ⊂ E m+1 s with pseudo-spherical Gauss map.…”