A differential 1-form on a $(2k+1)$-dimensional manifolds $M$ defines a singular contact structure if the set $S$ of points where the contact condition is not satisfied, $S=\{p\in M:(\omega \wedge {\left(d\omega \right)}^k\left(p\right)=0\}$, is nowhere dense in $M$. Then $S$ is a hypersurface with singularities and the restriction of $\omega$ to $S$ can be defined. Our first theorem states that in the holomorphic, real-analytic, and smooth categories the germ of Pfaffian equation $\left(\omega \right)$ generated by $\omega$ is determined, up to a diffeomorphism, by its restriction to $S$, if we eliminate certain degenerated singularities...

N. S. Sinyukov [5] introduced the concept of an almost geodesic mapping of a space $A_n$ with an affine connection without torsion onto ${\overline{A}}_n$ and found three types: ${\pi }_1$, ${\pi }_2$ and ${\pi }_3$. The authors of [1] proved completness of that classification for $n>5$.By definition, special types of mappings ${\pi }_1$ are characterized by equations $$P_{ij,k}^h+P_{ij}^{\alpha }{P}_{\alpha k}^h=a_{ij}{\delta }_k^h,$$ where $P_{ij}^h\equiv {\overline{\Gamma }}_{ij}^h-{\Gamma }_{ij}^h$ is the deformation tensor of affine connections of the spaces $A_n$ and ${\overline{A}}_n$.In this paper geometric objects which preserve these mappings are found and also closed classes of such spaces...

Starting by the famous paper by Kirillov, local Lie algebras of functions over smooth manifolds were studied very intensively by mathematicians and physicists. In the present paper we study local Lie algebras of pairs of functions which generate infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds.

L’espace de module des applications stables vers l’espace projectif possède naturellement une structure réelle dont la partie réelle est une variété projective normale. Cette dernière est un espace de module pour les courbes spatiales rationnelles réelles avec des points marqués réels. Puisque le lieu singulier est de codimension au moins deux, une première classe de Stiefel-Whitney est bien définie. Dans cet article nous déterminons un représentant pour la première classe de Stiefel-Whitney dans...

We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces.We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz...

Several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the symmetries of geometric structures and differential equations. A general result clarifying this effect in the case when the structure is associated to a vector distribution, is proposed.

Let $X$ be a uniruled projective manifold and let $x$ be a general point. The main result of [2] says that if the $(-K_X)$-degrees (i.e., the degrees with respect to the anti-canonical bundle of $X$) of all rational curves through $x$ are at least $dimX+1$, then $X$ is a projective space. In this paper, we study the structure of $X$ when the $(-K_X)$-degrees of all rational curves through $x$ are at least $dimX$. Our study uses the projective variety ${𝒞}_x\subset \mathbb{P}{T}_x\left(X\right)$, called the VMRT at $x$, defined as the union of tangent directions to the rational curves...