Related papers: Lines on contact manifolds II

Let C be a smooth complex projective curve of genus g and let X be its second symmetric product. This paper concerns the study of some attempts at extending to X the notion of gonality. In particular, we prove that the degree of…

The regular type of a real hyper-surface M in an (almost) complex manifold at some point p is the maximal contact order at p of M with germs of non singular (pseudo) holomorphic disks. The main purpose of this paper is to give two intrinsic…

Let f: X -> Y be a smooth family of canonically polarized complex varieties over a smooth base. Generalizing the classical Shafarevich hyperbolicity conjecture, Viehweg conjectured that Y is necessarily of log general type if the family has…

We construct an associative ring which is a deformation of the quantum cohomology ring of the projective plane. Just as the quantum cohomology encodes the incidence characteristic numbers of rational plane curves, the contact cohomology…

We obtain a sub-Riemannian version of the classical Gauss-Bonnet theorem. We consider subsurfaces of a three dimensional contact sub-Riemannian manifolds, and using a family of taming Riemannian metric, we obtain a pure sub-Riemannian…

Given a morphism between smooth projective varieties $f: W \to X$, we study whether $f$-relatively free rational curves imply the existence of $f$-relatively very free rational curves. The answer is shown to be positive when the fibers of…

We define a differential graded algebra associated to Legendrian knots in Seifert fibered spaces with transverse contact structures. This construction is distinguished from other combinatorial realizations of contact homology invariants by…

We study the geometry of the space of rational curves on smooth complete intersections of low degree, which pass through a given set of points on the variety. The argument uses spreading out to a finite field, together with an adaptation to…

Let f : Y -> X be a morphism of complex projective manifolds, and let F be a subsheaf of the tangent bundle which is closed under the Lie bracket, but not necessarily a foliation. This short paper contains an elementary and very geometric…

The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold $M$. The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component…

An irrational toric variety X is an analytic subset of the simplex associated to a finite configuration of real vectors. The positive torus acts on X by translation, and we consider limits of sequences of these translations. Our main result…

The canonical-type connection on the almost contact manifolds with B-metric is constructed. It is proved that its torsion is invariant with respect to a subgroup of the general conformal transformations of the almost contact B-metric…

Biharmonic or polyharmonic curves and surfaces in 3-dimensional contact manifolds are investigated.

We show that the dual of the cone of divisors on a complete Q-factorial toric variety X whose stable base loci have dimension less than k is generated by curves on small modifications that move in families sweeping out the birational…

Given a smooth hypersurface $X\subset \mathbb{P}^{n+1}$ of degree $d\geqslant 2$, we study the cones $V^h_p\subset \mathbb{P}^{n+1}$ swept out by lines having contact order $h\geqslant 2$ at a point $p\in X$. In particular, we prove that if…

A classification scheme of the conformal almost contact metric manifolds with respect to the covariant derivative of the Lee form is given. The subclasses of one basic class and their exact characterizations by the maximal subgroups of the…

Unlike Legendrian submanifolds, the deformation problem of coisotropic submanifolds can be obstructed. Starting from this observation, we single out in the contact setting the special class of integral coisotropic submanifolds as the direct…

An important class of contact 3--manifolds are those that arise as links of rational surface singularities with reduced fundamental cycle. We explicitly describe symplectic caps (concave fillings) of such contact 3--manifolds. As an…

The theory of linear transports along paths in vector bundles, generalizing the parallel transports generated by linear connections, is developed. The normal frames for them are defined as ones in which their matrices are the identity…

Contact homology for Legendrian submanifolds in standard contact $(2n+1)$-space is rigorously defined using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex $n$-space. It provides new invariants of…