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 note we show that any complete Kähler (immersed) Euclidean hypersurface $M^{2n}\subset {ℝ}^{2n+1}$ must be the product of a surface in ${ℝ}^3$ with an Euclidean factor ${ℂ}^{n-1}\cong {ℝ}^{2n-2}$.

We prove that a connected Riemannian manifold admitting a pair of non-trivial Einstein-Weyl structures $(g,\pm \omega )$ with constant scalar curvature is either Einstein, or the dual field of $\omega$ is Killing. Next, let $(M^n,g)$ be a complete and connected Riemannian manifold of dimension at least $3$ admitting a pair of Einstein-Weyl structures $(g,\pm \omega )$. Then the Einstein-Weyl vector field $E$ (dual to the $1$-form $\omega$) generates an infinitesimal harmonic transformation if and only if $E$ is Killing.

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...

On étudie la complétude géodésique des flots nul-prégéodésiques sur les variétés lorentziennes compactes, ce qui donne une obstruction à être nul-géodésique. On montre que lorsque l’orthogonal du champ de vecteurs engendrant le flot considéré s’intègre en un feuilletage $ℱ$, la complétude du flot se lit sur l’holonomie de $ℱ$. On montre ainsi qu’il n’existe pas de flots nul-géodésiques lisses sur $S^3$. On montre aussi qu’un $2$-tore lorentzien est nul-complet si et seulement si ses feuilletages de type lumière...

Holomorphic maps of Cartan domains of type four preserving the supports of complex geodesics are characterized, providing, in particular, a new description of holomorphic isometries.