We give a necessary and sufficient condition for a Codazzi structure to be realized as a minimal affine hypersurface or a minimal centroaffine immersion of codimension two.

A new class of $(n+1)$-dimensional Lorentz spaces of index $1$ is introduced which satisfies some geometric conditions and can be regarded as a generalization of Lorentz space form. Then, the compact space-like hypersurface with constant scalar curvature of this spaces is investigated and a gap theorem for the hypersurface is obtained.

A Lorentz surface of an indefinite space form is called a parallel surface if its second fundamental form is parallel with respect to the Van der Waerden-Bortolotti connection. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in general relativity since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel Lorentz surfaces in 4D neutral pseudo Euclidean...

In this paper we construct complete minimal surfaces of arbitrary genus in ${ℝ}^3$ with one, two, three and four ends respectively. Furthermore the surfaces lie between two parallel planes of ${ℝ}^3$.

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 ${\int }_{M}i{\left|A\right|}^{\alpha }<\infty$, where α ∈ [2(1 - √(2/n)), 2(1 + √(2/n))], then M is an affine n-dimensional plane. In particular, if n ≤ 8 and ${\int }_M{\left|A\right|}^d<\infty$, d = 1,3, then M is an affine n-dimensional plane. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite $L^{\alpha }$-norm curvature in ℝ⁷ are considered.

In this paper, we characterize the $n$-dimensional $(n\ge 3)$ complete spacelike hypersurfaces $M^n$ in a de Sitter space $S_1^{n+1}$ with constant scalar curvature and with two distinct principal curvatures one of which is simple.We show that $M^n$ is a locus of moving $(n-1)$-dimensional submanifold $M_1^{n-1}\left(s\right)$, along $M_1^{n-1}\left(s\right)$ the principal curvature $\lambda$ of multiplicity $n-1$ is constant and $M_1^{n-1}\left(s\right)$ is umbilical in $S_1^{n+1}$ and is contained in an $(n-1)$-dimensional sphere $S^{n-1}\left(c\left(s\right)\right)=E^n\left(s\right)\cap S_1^{n+1}$ and is of constant curvature ${\left(\frac{d\{log|{\lambda }^2-(1-R){|}^{1/n}\}}{ds}\right)}^2-{\lambda }^2+1$,where $s$ is the arc length of an orthogonal trajectory of the family...