arXiv subjects
Publications and source records attributed to Simon Vialaret.
In contact geometry, a systolic inequality is a uniform upper bound on the shortest period of a closed Reeb orbit, in terms of the contact volume. We prove a general systolic inequality valid on Seifert bundles with non-zero Euler number for all contact forms that are invariant under the underlying circle action.
The systole of a contact form $α$ is defined as the shortest period of closed Reeb orbits of $α$. Given a non-trivial $\mathbb S^1$-principal bundle over $\mathbb S^2$ with total space $M$, we prove a sharp systolic inequality for the class of tight contact form on $M$ invariant under the $\mathbb S^1$-action. This inequality exhibits a behavior which depends on the Euler class of the bundle in a subtle way. As applications, we prove a sharp systolic inequality for rotationally symmetric Finsler metrics on $\mathbb S^2$, a systolic inequality for the shortest contractible closed Reeb orbit, and a particular case of a conjecture by Viterbo.