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

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