On the remainder of the semialgebraic Stone–Cěch compactification of a semialgebraic set

In this work we analyze some topological properties of the remainder $?M:={\beta }_{\mathrm{s}}^{?}M?M$ of the semialgebraic Stone–Cěch compactification ${\beta }_{\mathrm{s}}^{?}M$ of a semialgebraic set $M?{\mathbb{R}}^m$ in order to ‘distinguish’ its points from those of M. To that end we prove that the set of points of ${\beta }_{\mathrm{s}}^{?}M$ that admit a metrizable neighborhood in ${\beta }_{\mathrm{s}}^{?}M$ equals $M_{\mathrm{lc}}\cup ({\mathrm{Cl}}_{{\beta }_{\mathrm{s}}^{?}M}({\stackrel{￣}{M}}_{\le 1})?{\stackrel{￣}{M}}_{\le 1})$ where $M_{\mathrm{lc}}$ is the largest locally compact dense subset of M and ${\stackrel{￣}{M}}_{\le 1}$ is the closure in M of the set of 1-dimensional points of M. In addition, we analyze the properties of the sets $\stackrel{?}{?}M$ and $\stackrel{?}{?}M$ of free maximal ideals associated with formal and semialgebraic paths. We prove that both are dense subsets of the remainder ?M and that the differences $?M?\stackrel{?}{?}M$ and $\stackrel{?}{?}M?\stackrel{?}{?}M$ are also dense subsets of ?M. It holds moreover that all the points of $\stackrel{?}{?}M$ have countable systems of neighborhoods in ${\beta }_{\mathrm{s}}^{?}M$.     

Minimal unit vector fields with respect to Riemannian natural metrics

Let $(M,g)$ be a Riemannian manifold. We denote by $\stackrel{?}{G}$ an arbitrary Riemannian g-natural metric on the unit tangent sphere bundle $T_{1}M$, such metric depends on four real parameters satisfying some inequalities. The Sasaki metric, the Cheeger–Gromoll metric and the Kaluza–Klein metrics are special Riemannian g-natural metrics. In literature, minimal unit vector fields have been already investigated, considering $T_{1}M$ equipped with the Sasaki metric ${\stackrel{?}{G}}_S$ [12]. In this paper we extend such characterization to an arbitrary Riemannian g-natural metric $\stackrel{?}{G}$. In particular, the minimality condition with respect to the Sasaki metric ${\stackrel{?}{G}}_S$ is invariant under a two-parameters deformation of the Sasaki metric. Moreover, we show that a minimal unit vector field (with respect to $\stackrel{?}{G}$) corresponds to a minimal submanifold. Then, we give examples of minimal unit vector fields (with respect to $\stackrel{?}{G}$). In particular, we get that the Hopf vector fields of the unit sphere, the Reeb vector field of a K-contact manifold, and the Hopf vector field of a quasi-umbilical hypersurface with constant principal curvatures in a Kähler manifold, are minimal unit vector fields (with respect to $\stackrel{?}{G}$).     

Biharmonic ideal hypersurfaces in Euclidean spaces

Let $x:M\to {\mathbb{E}}^m$ be an isometric immersion from a Riemannian n-manifold into a Euclidean m-space. Denote by Δ and $\overrightarrow{x}$ the Laplace operator and the position vector of M, respectively. Then M is called biharmonic if ${\mathrm{\Delta }}^2\overrightarrow{x}=0$. The following Chen?s Biharmonic Conjecture made in 1991 is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper we prove that the biharmonic conjecture is true for $\delta (2)$-ideal and $\delta (3)$-ideal hypersurfaces of a Euclidean space of arbitrary dimension.     

Bipartite algebraic graphs without quadrilaterals

Let ${\mathbb{P}}^s$ be the $s$-dimensional complex projective space, and let $X,Y$ be two non-empty open subsets of ${\mathbb{P}}^s$ in the Zariski topology. A hypersurface $H$ in ${\mathbb{P}}^s\times {\mathbb{P}}^s$ induces a bipartite graph $G$ as follows: the partite sets of $G$ are $X$ and $Y$, and the edge set is defined by $\overline{u}～\overline{v}$ if and only if $(\overline{u},\overline{v})\in H$. Motivated by the Turán problem for bipartite graphs, we say that $H\cap (X\times Y)$ is $(s,t)$-grid-free provided that $G$ contains no complete bipartite subgraph that has $s$ vertices in $X$ and $t$ vertices in $Y$. We conjecture that every $(s,t)$-grid-free hypersurface is equivalent, in a suitable sense, to a hypersurface whose degree in $\overline{y}$ is bounded by a constant $d=d(s,t)$, and we discuss possible notions of the equivalence.We establish the result that if $H\cap (X\times {\mathbb{P}}^2)$ is $(2,2)$-grid-free, then there exists $F\in \mathbb{C}[\overline{x},\overline{y}]$ of degree $\le 2$ in $\overline{y}$ such that $H\cap (X\times {\mathbb{P}}^2)=\{F=0\}\cap (X\times {\mathbb{P}}^2)$. Finally, we transfer the result to algebraically closed fields of large characteristic.     

Extended Picard complexes for algebraic groups and homogeneous spaces

For a smooth geometrically integral algebraic variety X over a field k of characteristic 0, we define the extended Picard complex $\mathrm{UPic}(\overline{X})$. It is a complex of length 2 which combines the Picard group $\mathrm{Pic}(\overline{X})$ and the group $U(\overline{X}):=\overline{k}{[\overline{X}]}^{\times }/{\overline{k}}^{\times }$, where $\overline{k}$ is a fixed algebraic closure of k and $\overline{X}=X{\times }_k\overline{k}$. For a connected linear k-group G we compute the complex $\mathrm{UPic}(\overline{G})$ (up to a quasi-isomorphism) in terms of the algebraic fundamental group ${\pi }_1(\overline{G})$. We obtain similar results for a homogeneous space X of a connected k-group G. To cite this article: M. Borovoi, J. van Hamel, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On boundaries of Levi-flat hypersurfaces in Cn

Let S be a smooth 2-codimensional real compact submanifold of ${\mathbb{C}}^n$, $n>2$. We address the problem of finding a compact hypersurface M, with boundary S, such that $M?S$ is Levi-flat. We prove the following theorem. Assume that (i) S is nonminimal at every CR point, (ii) every complex point of S is flat and elliptic and there exists at least one such point, (iii) S does not contain complex submanifolds of dimension $n?2$. Then there exists a Levi-flat $(2n?1)$-subvariety $\stackrel{?}{M}?\mathbb{C}\times {\mathbb{C}}^n$ with negligible singularities and boundary $\stackrel{?}{S}$ (in the sense of currents) such that the natural projection $\pi :\mathbb{C}\times {\mathbb{C}}^n\to {\mathbb{C}}^n$ restricts to a CR diffeomorphism between S and $\stackrel{?}{S}$. To cite this article: P. Dolbeault et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

On nice and injective-nice tournaments

One way of defining an oriented colouring of a directed graph $\overrightarrow{G}$ is as a homomorphism from $\overrightarrow{G}$ to a target directed graph $\overrightarrow{H}$, and an injective oriented colouring of $\overrightarrow{G}$ can be defined as a homomorphism from $\overrightarrow{G}$ to a target directed graph $\overrightarrow{H}$ such that no two in-neighbours of a vertex of $\overrightarrow{G}$ have the same image. Oriented colourings may be constructed using target directed graphs that are nice, as defined by Hell et al. (2001). We extend the work of Hell et al. by considering target graphs that are tournaments, characterizing nice tournaments, and proving that every nice tournament on $n$ vertices is $k$-nice for some $k\le n+2$. We also give a characterization of tournaments that are nice but not injective-nice.