We study the properties of the graph of a totally geodesic foliation. We limit our considerations to basic properties of the graph, and from them we derive several interesting corollaries on the structure of leaves.

We study some properties of the polar curve $P_l^{ℱ}$ associated to a singular holomorphic foliation $ℱ$ on the complex projective plane ${\mathbb{P}}^2$. We prove that, for a generic center $l\in {\mathbb{P}}^2$, the curve $P_l^{ℱ}$ is irreducible and its singular points are exactly the singular points of $ℱ$ with vanishing linear part. We also obtain upper bounds for the algebraic multiplicities of the singularities of $ℱ$ and for its number of radial singularities.

Soit $(M,ℱ)$ un feuilletage riemannien sur une variété compacte; $\overline{ℱ}$ est le feuilletage singulier défini par les adhérences des feuilles $(\bar{F},ℱ)$ le feuilletage induit sur une adhérence générique. On étudie le cas où $(\bar{F},ℱ)$ n’a pas de champ transverse non trivial. Alors l’espace quotient $W=M/\overline{ℱ}$ a une structure naturelle de variété de Sataké, de manière que la projection $M\to W$ soit un morphisme (de variétés de Sataké) avec pliage autour des adhérences singulières.

It is known that, for a regular riemannian foliation on a compact manifold, the properties of its basic cohomology (non-vanishing of the top-dimensional group and Poincaré duality) and the tautness of the foliation are closely related. If we consider singular riemannian foliations, there is little or no relation between these properties. We present an example of a singular isometric flow for which the top-dimensional basic cohomology group is non-trivial, but the basic cohomology does not satisfy...

Soient $X$ une variété de Hadamard de courbure $\le -1$ et $\Gamma$ un groupe d’isométries non élémentaire. Nous montrons qu’il y a équivalence entre la non-arithméticité du spectre des longueurs de $\Gamma \setminus X$, le mélange topologique du flot géodésique et l’existence d’une feuille dense pour le feuilletage fortement stable.

In this paper we study a curvature integral associated with a pair of orthogonal foliations on the Riemann sphere S 2 and we compute the minimal value of the volume of meromorphic foliations.

Consider two foliations ${ℱ}_1$ and ${ℱ}_2$, of dimension one and codimension one respectively, on a compact connected affine manifold $(M,\nabla )$. Suppose that ${\nabla }_{T{ℱ}_1}T{ℱ}_2\subset T{ℱ}_2$; ${\nabla }_{T{ℱ}_2}T{ℱ}_1\subset T{ℱ}_1$ and $TM=T{ℱ}_1\oplus T{ℱ}_2$. In this paper we show that either ${ℱ}_2$ is given by a fibration over $S^1$, and then ${ℱ}_1$ has a great degree of freedom, or the trace of ${ℱ}_1$ is given by a few number of types of curves which are completely described. Moreover we prove that ${ℱ}_2$ has a transverse affine structure.

In this paper, we prove that the composition of a transversal biwave map and a transversally totally geodesic map is a transversal biwave map. We show that there are biwave maps which are not transversal biwave maps, and there are transversal biwave maps which are not biwave maps either. We prove that if $f$ is a transversal biwave map satisfying certain condition, then $f$ is a transversal wave map. We finally study the transversal conservation laws of transversal biwave maps.

We establish in this paper a lower bound for the volume of a unit vector field $\overrightarrow{v}$ defined on ${𝐒}^n\setminus \{\pm x\}$, $n=2,3$. This lower bound is related to the sum of the absolute values of the indices of $\overrightarrow{v}$ at $x$ and $-x$.