1. CJM 2012 (vol 65 pp. 621)
|On Surfaces in Three Dimensional Contact Manifolds|
In this paper, we introduce two notions on a surface in a contact manifold. The first one is called degree of transversality (DOT) which measures the transversality between the tangent spaces of a surface and the contact planes. The second quantity, called curvature of transversality (COT), is designed to give a comparison principle for DOT along characteristic curves under bounds on COT. In particular, this gives estimates on lengths of characteristic curves assuming COT is bounded below by a positive constant.
We show that surfaces with constant COT exist and we classify all graphs in the Heisenberg group with vanishing COT. This is accomplished by showing that the equation for graphs with zero COT can be decomposed into two first order PDEs, one of which is the backward invisicid Burgers' equation. Finally we show that the p-minimal graph equation in the Heisenberg group also has such a decomposition. Moreover, we can use this decomposition to write down an explicit formula of a solution near a regular point.
Keywords:contact manifolds, subriemannian manifolds, surfaces
2. CJM 2005 (vol 57 pp. 1314)
|Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra |
In 1999 V. Arnol'd introduced the local contact algebra: studying the problem of classification of singular curves in a contact space, he showed the existence of the ghost of the contact structure (invariants which are not related to the induced structure on the curve). Our main result implies that the only reason for existence of the local contact algebra and the ghost is the difference between the geometric and (defined in this paper) algebraic restriction of a $1$-form to a singular submanifold. We prove that a germ of any subset $N$ of a contact manifold is well defined, up to contactomorphisms, by the algebraic restriction to $N$ of the contact structure. This is a generalization of the Darboux-Givental' theorem for smooth submanifolds of a contact manifold. Studying the difference between the geometric and the algebraic restrictions gives a powerful tool for classification of stratified submanifolds of a contact manifold. This is illustrated by complete solution of three classification problems, including a simple explanation of V.~Arnold's results and further classification results for singular curves in a contact space. We also prove several results on the external geometry of a singular submanifold $N$ in terms of the algebraic restriction of the contact structure to $N$. In particular, the algebraic restriction is zero if and only if $N$ is contained in a smooth Legendrian submanifold of $M$.
Keywords:contact manifold, local contact algebra,, relative Darboux theorem, integral curves
Categories:53D10, 14B05, 58K50
3. CJM 1999 (vol 51 pp. 1123)
|First Steps of Local Contact Algebra |
We consider germs of mappings of a line to contact space and classify the first simple singularities up to the action of contactomorphisms in the target space and diffeomorphisms of the line. Even in these first cases there arises a new interesting interaction of local commutative algebra with contact structure.
Keywords:contact manifolds, local contact algebra, Diracian, contactian