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IMRN International Mathematics Research Notices 1999, No. 4

Summary: IMRN International Mathematics Research Notices
1999, No. 4
A Note on V. I. Arnold's Chord Conjecture
Casim Abbas
1 Introduction
This paper makes a contribution to a conjecture of V. I. Arnold in contact geometry, which
he stated in his 1986 paper [4]. A [4] on a closed, oriented three manifold M is a 1-form
so that d is a volume form. There is a distinguished vectorfield X, called the Reeb
vectorfield of , which is defined by iX d 0 and iX 1. The standard example on S3
is the following: Consider the 1-form on R4
defined by
(x1dy1 - y1dx1 + x2dy2 - y2dx2).
This induces a contact form on the unit three sphere S3
. Observe that all the orbits of
the Reeb vectorfield are periodic; they are the fibres of the Hopf fibration. Note that the
dynamics of the Reeb vectorfield changes drastically in general if we replace by the
contact form f where f is a nowhere vanishing function on S3


Source: Abbas, Casim - Department of Mathematics, Michigan State University


Collections: Mathematics