J-holomorphic Curves, Legendre Submanifolds and Reeb Chords

Floer homology for negative line bundles and Reeb chords in pre-quantization spaces

In this article we prove existence of Reeb orbits for Bohr-Sommerfeld Legendrians in certain pre-quantization spaces. We give a quantitative estimate from below. These estimates are obtained by studying Floer homology for fibre-wise quadratic Hamiltonian functions on negative line bundles.

Healthy vector spaces and spicy Hopf algebras (with applications to the growth rate of geodesic chords and to intermediate volume growth on manifolds of non-finite type)

We give lower bounds for the growth of the number of Reeb chords and for the volume growth of Reeb flows on spherizations over closed manifolds M that are not of finite type, have virtually polycyclic fundamental group, and satisfy a mild assumption on the homology of the based loop space. For the special case of geodesic flows, these lower bounds are: (i) For any Riemannian metric on M, any pair of non-conjugate points p,q in M, and every component C of the space of paths from p to q, the number of geodesics in C of length at most T grows at least like e^{\sqrt T}. (ii) The exponent of the vo…

N-Legendre and N-Slant Curves in the Unit Tangent Bundle of Minkowski Surfaces

Let $(\mathbb{M}_{1}^{2},g)$ be a Minkowski surface and $(T_1\mathbb{M}_1^2, g_1)$ its unit tangent bundle endowed with the pseudo-Riemannian induced Sasaki metric. We extend in this paper the study of the N-Legendre and N-slant curves which the inner product of normal vector and Reeb vector is zero and nonzero constant respectively in $\left( T_1 \mathbb{M}_1^2, g_1 \right)$, given in \cite{hmy}, to the Minkowski context and several important characterizations of these curves are given.\newline

Bi-paracontact structures and Legendre foliations

We study almost bi-paracontact structures on contact manifolds. We prove that if an almost bi-paracontact structure is defined on a contact manifold $(M,η)$, then under some natural assumptions of integrability, $M$ carries two transverse bi-Legendrian structures. Conversely, if two transverse bi-Legendrian structures are defined on a contact manifold, then $M$ admits an almost bi-paracontact structure. We define a canonical connection on an almost bi-paracontact manifold and we study its curvature properties, which resemble those of the Obata connection of an anti-hypercomplex (or complex-pro…

Explicit formulas for biharmonic submanifolds in Sasakian space forms

We classify the biharmonic Legendre curves in a Sasakian space form, and obtain their explicit parametric equations in the $(2n+1)$-dimensional unit sphere endowed with the canonical and deformed Sasakian structures defined by Tanno. Then, composing with the flow of the Reeb vector field, we transform a biharmonic integral submanifold into a biharmonic anti-invariant submanifold. Using this method we obtain new examples of biharmonic submanifolds in spheres and, in particular, in $\mathbb{S}^{7}$.

Sasakian Lorentzian 3-维 manifold 中的接触涡线

Contact Whirl Curves in Sasakian Lorentzian 3-Manifolds

本文研究了Sasakian Lorentzian 3-维 manifold 中的接触涡线，通过Reeb向量场编码曲线适应框架与环境接触结构的相互作用，推导了非测地单位速度接触涡线的微分方程，并证明了非测地Legendre Frenet曲线具有常数 torsion τ=1。