We study non-totally geodesic Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms of the so-called angle functions and that such Lagrangian submanifolds are closely related to minimal surfaces in 𝕊3. Indeed, starting from an arbitrary minimal surface, we can construct locally a large family of such Lagrangian immersions, including one exceptional example. We also show that locally all such Lagrangian submanifolds can be obtained in this way.

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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...

Abstract.In this work, we focus on a class of timelike rotational surfaces in Minkowski space E 4 1 with 2-dimensional axis. There are three types of rotational surfaces with 2-dimensional axis, called rotational surfaces of elliptic, hyperbolic or parabolic type. We obtain all flat timelike rotational surface of elliptic and hyperbolic types with pointwise 1-type Gauss map of the first and second kind. We also prove that there exists no flat timelike rotational surface of parabolic type in E 4 1 with pointwise 1-type Gauss map.

Chen and Lue (2007) initiated the study of spherical submanifolds with nite type spherical Gauss map. In this paper, we rstly prove that a submanifold M n of the unit sphere m− has non-mass-symmetric -type spherical Gauss map if and only if M n is an open part of a small n-sphere of a totally geodesic (n + )-sphere n+ ⊂ m− . Then we show that a non-totally umbilical hypersurface M of n+ with nonzero constant mean curvature in n+ has mass-symmetric -type spherical Gauss map if and only if the scalar curvature curvature of M is constant. Finally, we classify constant mean curvature surfaces in with mass-symmetric -type spherical Gauss map.

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Abstract. In this work, we study two families of rotational surfaces in the pseudo-Euclidean space E 4 2 with profile curves lying in 2-dimensional planes. First, we obtain a classification of pseudo-umbilical spacelike surfaces and timelike surfaces in these families. Then, we show that in this classification there exists no a pseudo-umbilical rotational surface in E 4 2 with pointwise 1-type Gauss map of second kind. Finally, we determine such pseudo-umbilical rotational surfaces in E 4 2 having pointwise 1-type Gauss map of first kind.