Generalized constant ratio hypersurfaces in Euclidean spaces

In this paper, we study generalized constant ratio (GCR) hypersurfaces in Euclidean spaces. We mainly focus on the hypersurfaces in $\mathbb E^4$. First, we deal with $\delta(2)$-ideal GCR hypersurfaces. Then, we study on hypersurfaces with constant (first) mean curvature. Finally, we obtain the complete classification of GCR hypersurfaces with vanishing Gauss-Kronecker curvature. We also give some explicit examples. Keywords: Generalized constant ratio submanifolds, $\delta(r)$-invariant hypersurfaces, constant mean curvature, Gauss-Kronecker curvatur

On the quasi-minimal surfaces in the 4-dimensional de Sitter space with 1-type Gauss map

In this paper, we study the Gauss map of the surfaces in the de Sitter space-time $\mathbb S^4_1(1)$. First, we prove that a space-like surface lying in the de Sitter space-time has pointwise 1-type Gauss map if and only if it has parallel mean curvature vector. Then, we obtain the complete classification of the quasi-minimal surfaces with 1-type Gauss map.Comment: This work has been presented in 2nd International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2013

On the marginally trapped surfaces in Minkowski space-time with finite type Gauss map

In this paper, we work on the marginally trapped surfaces in the 4-dimensional Minkowski, de Sitter and anti-de Sitter space-times. We obtain the complete classification of the marginally trapped surfaces in the Minkowski space-time with pointwise 1-type Gauss map. Further, we give construction of marginally trapped surfaces with 1-type Gauss map and a given boundary curve. We also state some explicit examples. We also prove that a marginally trapped surface in the de Sitter space-time $\mathbb S^4_1(1)$ or anti-de Sitter space-time $\mathbb H^4_1(-1)$ has pointwise 1-type Gauss map if and only if its mean curvature vector is parallel. Moreover, we obtain that there exists no marginally trapped surface in $\mathbb S^4_1(1)$ or $\mathbb H^4_1(-1)$ with harmonic Gauss map

H-hypersurfaces with at most 3 distinct principal curvatures in the Euclidean spaces

In this paper, we study hypersurfaces of Euclidean spaces with arbitrary dimension. First, we obtain some results on \mbox{H}-hypersurfaces. Then, we give the complete classification of \mbox{H}-hypersurfaces with 3 distinct curvatures. We also give explicit examples

On the Lorentzian minimal surfaces in E14\mathbb E^4_1 with finite type Gauss map

In this paper, we study the Lorentzian minimal surfaces in the Minkowski space-time with finite type Gauss map. First, we obtain the classification of this type of surfaces with pointwise 1-type Gauss map. Then, we proved that there are no Lorentzian minimal surface in the Minkowski space-time with null 2-type Gauss map

Classification of minimal Lorentzian surfaces in S24(1)\mathbb S^4_2(1) with constant Gaussian and normal curvatures

In this paper we consider Lorentzian surfaces in the 4-dimensional pseudo-Riemannian sphere $\mathbb S^4_2(1)$ with index 2 of curvature one. We obtain the complete classification of minimal Lorentzian surfaces $\mathbb S^4_2(1)$ whose Gaussian and normal curvatures are constants. We conclude that such surfaces have the Gaussian curvature $1/3$ and the absolute value of normal curvature $2/3$. We also give some explicit examples.Comment: Keywords. Gaussian curvature, minimal submanifolds, Lorentzian surfaces, normal curvatur

A Classification of Biconservative Hypersurfaces in a Pseudo-Euclidean Space

In this paper, we study biconservative hypersurfaces of index 2 in $\mathbb E^{5}_{2}$. We give the complete classification of biconservative hypersurfaces with diagonalizable shape operator at exactly three distinct principal curvatures. We also give an explicit example of biconservative hypersurfaces with four distinct principal curvatures

Space-like Surfaces in Minkowski Space E14\mathbb E^4_1 with Pointwise 1-Type Gauss Map

In this work we firstly classify space-like surfaces in Minkowski space $\mathbb E^4_1$, de-Sitter space $\mathbb S^3_1$ and hyperbolic space $\mathbb H^3$ with harmonic Gauss map. Then we give a characterization and classification of space-like surfaces with pointwise 1-type Gauss map of the first kind. We also give some explicit examples.Comment: arXiv admin note: text overlap with arXiv:1302.2910 by other author

Complete classification of biconservative hypersurfaces with diagonalizable shape operator in Minkowski 4-space

In this paper, we study biconservative hypersurfaces in the four dimensional Minkowski space $\mathbb E^4_1$. We give the complete explicit classification of biconservative hypersurfaces with diagonalizable shape operator in $\mathbb E^4_1$.Comment: Keywords: Biharmonic submanifolds, Biconservative hypersurfaces, Minkowski space, Diagonalizable shape operato

Quasi-minimal Lorentz Surfaces with Pointwise 1-type Gauss Map in Pseudo-Euclidean 4-Space

A Lorentz surface in the four-dimensional pseudo-Euclidean space with neutral metric is called quasi-minimal if its mean curvature vector is lightlike at each point. In the present paper we obtain the complete classification of quasi-minimal Lorentz surfaces with pointwise 1-type Gauss map.Comment: 16 page