Related papers: On local geometry of rank 3 distributions with 6-d…

In this article, we study abnormal curves in a family of sub-Riemannian manifolds of rank 2. We focus on abnormal curves whose lifts to the cotangent bundle annihilate, at an interior point of the domain, all Lie brackets of length up to…

We prove a splitting theorem for Riemannian n-manifolds with scalar curvature bounded below by a negative constant and containing certain area-minimising hypersurfaces (Theorem 3). Thus we generalise [25,Theorem 3] by Nunes. This splitting…

We consider a special class of Finsler metrics --- square metrics which are defined by a Riemannian metric and a 1-form on a manifold. We show that an analogue of the Beltrami Theorem in Riemannian geometry is still true for square metrics…

A space curve in a Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf…

This paper deals with the problem of detecting non-isotropic high-dimensional geometric structure in random graphs. Namely, we study a model of a random geometric graph in which vertices correspond to points generated randomly and…

Let $X$ be a smooth projective variety. We show that the map that sends a codimension one distribution on $X$ to its singular scheme is a morphism from the moduli space of distributions into a Hilbert scheme. We describe its fibers and,…

We classify all possible charge-3 monopole spectral curves with non-trivial automorphism group and within these identify those with elliptic quotients. By focussing on elliptic quotients the transcendental constraints for a monopole…

We develop an algebraic version of Cartan method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space $W$ with respect to the action of a subgroup $G$ of the $GL(W)$. Under…

If $L$ is a semisimple Lie algebra of vector fields on R^N with a split Cartan subalgebra C, then it is proved that the dimension of the generic orbit of C coincides with the dimension of C. As a consequence one obtains a local canonical…

Based on the ideas of Optimal Control, we introduce the new basic characteristic of a bracket generating distribution, the Jacobi symbol. In contrast to the classical Tanaka symbol, the set of Jacobi symbols is discrete and classifiable. We…

We consider algebraic manifolds $Y$ of dimension 3 over $\Bbb{C}$ with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$ and $i>0$. Let $X$ be a smooth completion of $Y$ with $D=X-Y$, an effective divisor on $X$ with normal crossings. If the…

We classify six-dimensional homogeneous nearly K\"{a}hler manifolds and give a positive answer to Gray and Wolf's conjecture: every homogeneous nearly K\"{a}hler manifold is a Riemannian 3-symmetric space equipped with its canonical almost…

Complementing the previous paper in the series, this paper classifies $|2|$-graded parabolic geometries, listing their important properties: the group $G_0$, the graded tangent bundle $gr(T)$ and its algebra\"ic bracket, the relevant…

We develop a general theory for irreducible homogeneous spaces $M= G/H$, in relation to the nullity $\nu$ of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that…

We construct canonical frames and find all maximally symmetric models for a natural generic class of corank 2 distributions on manifolds of odd dimension greater or equal to 7. This class of distributions is characterized by the following…

The notion of a parallelizable distribution has been introduced and investigated. A non-integrable parallelizable distribution carries a natural sub-Riemannian structure. The geometry of this structure has been studied from the bi-viewpoint…

In this article we investigate the regularity properties of linear degenerations of flag varieties. We classify the linear degenerations of (partial) flag varieties that are smooth. Furthermore, we study the singular locus of irreducible…

For any $n\geq 3$, we prove that there exist equivalences between these apparently unrelated objects: irreducible $n$-dimensional non degenerate projective varieties $X\subset \mathbb P^{2n+1}$ different from rational normal scrolls and…

The aim of this paper is to verify that the study of generic conformally flat hypersurfaces in 4-dimensional space forms is reduced to a surface theory in the standard 3-sphere. The conformal structure of generic conformally flat…

We use algebraic geometry to study the anomaly-free representations of an arbitrary gauge Lie algebra for 4-dimensional spacetime fermions. For irreducible representations, the problem reduces to studying the Lie algebras $\mathfrak{su}_n$…