Sharp systolic inequalities for invariant tight contact forms on principal S1-bundles over S2

The systole of a contact form $\alpha$ is defined as the shortest period of closed Reeb orbits of $\alpha$. 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.