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Commuting Conditions of the k-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms

Konstantina Panagiotidou, Juan de Dios Pérez (2015)

Open Mathematics

In this paper three dimensional real hypersurfaces in non-flat complex space forms whose k-th Cho operator with respect to the structure vector field ξ commutes with the structure Jacobi operator are classified. Furthermore, it is proved that the only three dimensional real hypersurfaces in non-flat complex space forms, whose k-th Cho operator with respect to any vector field X orthogonal to structure vector field commutes with the structure Jacobi operator, are the ruled ones. Finally, results...

Compact hypersurfaces with constant higher order mean curvatures.

Antonio Ros Mulero (1987)

Revista Matemática Iberoamericana

A fundamental question about hypersurfaces in the Euclidean space is to decide if the sphere is the only compact hypersurface (embedded or immersed) with constant higher order mean curvature Hr, for some r = 1, ..., n.

Complete classification of spatial surfaces with parallel mean curvature vector in arbitrary non-flat pseudo-Riemannian space forms

Bang-Yen Chen (2009)

Open Mathematics

Submanifolds with parallel mean curvature vector play important roles in differential geometry, theory of harmonic maps as well as in physics. Spatial surfaces in 4D Lorentzian space forms with parallel mean curvature vector were classified by B. Y. Chen and J. Van der Veken in [9]. Recently, spatial surfaces with parallel mean curvature vector in arbitrary pseudo-Euclidean spaces are also classified in [7]. In this article, we classify spatial surfaces with parallel mean curvature vector in pseudo-Riemannian...

Complete noncompact submanifolds with flat normal bundle

Hai-Ping Fu (2016)

Annales Polonici Mathematici

Let Mⁿ (n ≥ 3) be an n-dimensional complete super stable minimal submanifold in ℝ n + p with flat normal bundle. We prove that if the second fundamental form A of M satisfies ∫ M i | A | α < ∞ , where α ∈ [2(1 - √(2/n)), 2(1 + √(2/n))], then M is an affine n-dimensional plane. In particular, if n ≤ 8 and ∫ M | A | d < ∞ , d = 1,3, then M is an affine n-dimensional plane. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L α -norm curvature in ℝ⁷ are considered.

Complete real Kähler Euclidean hypersurfaces are cylinders

Luis A. Florit, Fangyang Zheng (2007)

Annales de l’institut Fourier

In this note we show that any complete Kähler (immersed) Euclidean hypersurface M 2 n ⊂ ℝ 2 n + 1 must be the product of a surface in ℝ 3 with an Euclidean factor ℂ n - 1 ≅ ℝ 2 n - 2 .

Conformal nullity of isotropic submanifolds

Vladimir Rovenski (2005)

Annales Polonici Mathematici

We introduce and study submanifolds with extrinsic curvature and second fundamental form related by an inequality that holds for isotropic submanifolds and becomes equality for totally umbilical submanifolds. The dimension of umbilical subspaces and the index of conformal nullity of these submanifolds with low codimension are estimated from below. The corollaries are characterizations of extrinsic spheres in Riemannian spaces of positive curvature.

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