Hypersurfaces of a Lorentz-Minkowski space L n+1 with pointwise 1-type Gauss map are characterized. We prove that an oriented hypersurface M q in L n+1 has pointwise 1-type Gauss map of the first kind if and only if M q has constant mean curvature and conclude that all oriented isoparametric hypersurfaces in L n+1 have 1-type Gauss map. Then we classify rational rotation hypersurfaces of L n+1 with pointwise 1-type Gauss map and give some examples.

In this paper we prove that an oriented hypersurface M of a Euclidean space E n+1 has pointwise 1-type Gauss map of the first kind if and only if M has constant mean curvature. Then we conclude that all oriented isoparametric hypersurfaces of E n+1 has 1-type Gauss map. We also show that a rational hypersurface of revolution in a Euclidean space E n+1 has pointwise 1-type Gauss map of the second kind if and only if it is a right n-cone.

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In this work, we study the pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss map. First, we classify the Lorentzian surfaces in a 4-dimensional pseudo-sphere S 4 s (1) with index s, s = 1, 2, and having harmonic pseudo-spherical Gauss map. Then we give a characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere S m−1 s (1) ⊂ E m s with 1-type pseudospherical Gauss map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space S 4 1 (1) ⊂ E 5 1 with 1-type pseudo-spherical Gauss map. Finally, according to the causal character of the mean curvature vector we obtain the classification of submanifolds of a pseudo-sphere having 1-type pseudo-spherical Gauss map with nonzero constant component in its spectral decomposition 2010 Mathematics Subject Classification: Primary 53C40, 53C42, 53B25.Key words and phrases: finite type mapping, Gauss map, pseudo-sphere, biharmonic Gauss map, marginally trapped surface.where φ 0 is a constant vector in E m , and φ i 's are nonconstant E m -valued maps such thatWhen the spectral resolution contains exactly k nonconstant terms, the map φ is called of k-type. Otherwise, it is of infinite type. Since M is compact, φ 0 is the center of mass of M in E m . On the other hand, one cannot make the spectral decomposition of a map on a noncompact Riemannian manifold in general.However, the notion of finite type immersions on noncompact manifolds was given as above in [4, page 124]. For the noncompact case the Laplacian operator may have zero eigenvalue for a nonconstant map in the spectral decomposition. Let x : M → E m s be an isometric immersion from an n-dimensional pseudo-Riemannian manifold M into a pseudo-Euclidean space E m s . Let G(n, m) denote the Grassmannian manifold consisting of all oriented n-planes of E m s . The classical Gauss map ν : M → G(n, m) associated with x is the map which carries each point p ∈ M to the oriented n-plane of E m s obtained by parallel displacement of the tangent space T p M to the origin of E m s . Since G(n, m) can be canonically imbedded in the vector space n E m s ∼ = E N q for some integer q, the classical Gauss map ν gives rise to a well-defined map from M into the pseudo-Euclidean space E N q , where N = m n and n E m s is the vector space obtained by the exterior products of n vectors in E m s , [15]. An isometric immersion from an n-dimensional Riemannian manifold M into a Euclidean (m−1)-sphere S m−1 can be viewed as one into a Euclidean m-space, and therefore the Gauss map associated with such an immersion can be determined in the ordinary sense. However, for the Gauss map to reflect the properties of the immersion into a sphere, instead of into the Euclidean space, M. Obata modified the definition of Gauss map appropriately, [17]. Let x : M → M be an isometric immersion from an n-dimensional Riemannian manifold M into an m-dimensional simply connected complete space M of constant curvature. The generalized Gauss map in the Obata's sense is a map which assigns to...