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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 Category:35R03 |
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 Categories:53D10, 14B05 |