Associated Metrics for Bourgeois’ Contact Forms

Abstract

In 2002, Frédéric Bourgeois [2] showed that, given a compact contact manifold M 2n–1, the product M 2n–1 × T 2 also carries a contact form; then, as a corollary he observed that all odd-dimensional tori have contact forms. The idea is to use an open book decomposition of M 2n–1 that is compatible with its contact structure [4] to produce contact forms on the product M 2n–1 × T 2. Here we first find the Reeb vector field and then a method for constructing associated metrics for contact forms of the type studied by Bourgeois. We will also discuss S 2 × T 2 in some detail. While the procedure would apply to tori, the construction is quite difficult and we will make only some remarks.

Keywords

• mathematical modeling
• applied mathematics
• nicole oresme
• 14th century