On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds

Given a surface S in a 3D contact sub-Riemannian manifold M , we investigate the metric structure induced on S by M , in the sense of length spaces. First, we define a coefficient [[EQUATION]] at characteristic points that determines locally the characteristic foliation of S . Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points.

Transverse Kähler holonomy in Sasaki Geometry and S-Stability

We study the transverse Kähler holonomy groups on Sasaki manifolds (M, S) and their stability properties under transverse holomorphic deformations of the characteristic foliation by the Reeb vector field. In particular, we prove that when the first Betti number b 1(M) and the basic Hodge number h 0,2 B(S) vanish, then S is stable under deformations of the transverse Kähler flow. In addition we show that an irreducible transverse hyperkähler Sasakian structure is S-unstable, whereas, an irreducible transverse Calabi-Yau Sasakian structure is S-stable when dim M ≥ 7. Finally, we prove that the standard Sasaki join operation (transverse holonomy U(n 1) × U(n 2)) as well as the fiber join operation preserve S-stability.

Riemannian Lie Subalgebroid

This paper  talks about Riemannian Lie subalgebroid. We investigate the induced Levi-civita connection on  Riemannian Lie subalgebroid, and give a construction of the second fondamental form like in case of Riemannian submanifold. We also give the Gauss formula in the case of Riemannian Lie subalgebroid. In the case of the Lie subalgebroid induced by a Leaf of a characteristic foliation, we obtain that the leaf carries more curvature than the manifold as shown by  Boucetta (2011).

Deformations of Pre-symplectic Structures and the Koszul L∞-algebra

We study the deformation theory of pre-symplectic structures, that is, closed 2-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an $L_{\infty }$-algebra, which we call the Koszul $L_{\infty }$-algebra. This $L_{\infty }$-algebra is a cousin of the Koszul dg Lie algebra associated to a Poisson manifold. In addition, we show that a quotient of the Koszul $L_{\infty }$-algebra is isomorphic to the $L_{\infty }$-algebra that controls the deformations of the underlying characteristic foliation. Finally, we show that the infinitesimal deformations of pre-symplectic structures and of foliations are both obstructed.

Characteristic foliation on a hypersurface of general type in a projective symplectic manifold

A foliation on a non-singular projective variety is algebraically integrable if all leaves are algebraic subvarieties. A non-singular hypersurface X in a non-singular projective variety M equipped with a symplectic form has a naturally defined foliation, called the characteristic foliation on X. We show that if X is of general type and dim M≥4, then the characteristic foliation on X cannot be algebraically integrable. This is a consequence of a more general result on Iitaka dimensions of certain invertible sheaves associated with algebraically integrable foliations by curves. The latter is proved using the positivity of direct image sheaves associated to families of curves.

Rigidity properties of Anosov optical hypersurfaces

We consider an optical hypersurface Σ in the cotangent bundle τ:T*M→M of a closed manifold M endowed with a twisted symplectic structure. We show that if the characteristic foliation of Σ is Anosov, then a smooth 1-form θ on M is exact if and only if τ*θ has zero integral over every closed characteristic of Σ. This result is derived from a related theorem about magnetic flows which generalizes our previous work [N. S. Dairbekov and G. P. Paternain. Longitudinal KAM cocycles and action spectra of magnetic flows. Math. Res. Lett.12 (2005), 719–729]. Other rigidity issues are also discussed.

Formes normales d'équations differéntielles implicites et de champs de Liouville

Consider the partial differential equation f(x, y(x), dy(x)) = 0, where f is a smooth real function on ℝn × ℝ × (ℝn)*. Near each singularity of the characteristic foliation, a Liouville field is associated to the equation; we classify hyperbolic germs of Liouville fields up to symplectic transformations, hence we deduce normal forms for partial differential equations up to transformations which preserve the standard contact form of ℝ2n+1. For n = 1, a theorem of Davydov enables us to deduce normal forms for such equations up to transformations of the x, y plane.