Pseudo-Spherical Submanifolds with 1-Type Pseudo-Spherical Gauss Map

Abstract

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...

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

“…Consider a spacelike (i.e. Riemannian) surface with arbitrary codimension in a pseudo-sphere S m s (1). For the Riemannian case s = 0, we refer to [7].…”

“…In [1], the first author, Canfes and Dursun introduced the notion of pseudo-spherical Gauss map associated with an immersion of a (pseudo-)Riemannian manifold into a pseudo-sphere and also obtained some characterization and classification theorems. Note however that, in [13], Ishihara studied the Gauss map in a generalized sense for (pseudo-)Riemannian submanifolds of (pseudo-) Riemannian manifolds, also extending the Gauss map in Obata's sense to the pseudo-Riemannian setting.…”

Classification of surfaces in a pseudo‐sphere with 2‐type pseudo‐spherical Gauss map

“…Then, from (4.15) = 0. Also,̂= ( ± 1) 3 and ‖ ‖ ‖ĥ ‖ ‖ ‖ 2 = 2( ± 1) 2 . Since the mean curvature is non-zero constant, is constant and ≠ ±1.…”

“…In , we examined the pseudo‐Riemannian submanifolds in a pseudo‐sphere with 1‐type pseudo‐spherical Gauss map. We gave a complete classification of spacelike surfaces and Lorentzian surfaces in the de Sitter space with 1‐type pseudo‐spherical Gauss map.…”

Pseudo‐hyperbolic Gauss maps of Lorentzian surfaces in anti‐de Sitter space

“…the pseudo‐hyperbolic) Gauss maps of isometric immersions into the pseudo‐sphere (resp. the pseudo‐hyperbolic space) (see [2–4, 18]).…”