The aim of this paper is to prove a contact non-squeezing result for a class of embeddings between star-shaped domains in the contactization of the symplectization of the unit cotangent bundle of certain manifolds. The class of embeddings includes embeddings which are not isotopic to the identity. This yields a new proof that there is no positive loop of contactomorphisms in the unit cotangent bundles under consideration.
The arguments are inspired by the shape invariant introduced by J. C. Sikorav [Mém. Soc. Math. Fr., Nouv. Sér. 46, 151–167 (1991; Zbl 0751.58010)] and Y. Eliashberg [J. Am. Math. Soc. 4, No. 3, 513–520 (1991; Zbl 0733.58011)]. The main result of the paper is a non-squeezing theorem. Let \(B^n\), \(n>1\), be a compact and connected smooth manifold, let \(Y\) be the unit cotangent bundle of \(B\), and let \[ Q=\mathbb{R}/\mathbb{Z}\times SY, \] where \(SY\subset T^*B\) is the complement of the zero section. The radial coordinate function \(\rho:T^*B\longrightarrow[0,\infty)\), that is, the distance to the zero-section, determines a star-shaped domain \[ \Omega(r)=\{(t,z)\in Q:0<\rho(z)\leq r^2\}. \] The author proves the following: If \(B\) admits a closed one-form \(\beta\) such that \(\rho(\beta)=1\) everywhere, then there is no generalized squeezing of \(\Omega(R)\) into \(\Omega(r)\) for \(r<R\). The argument used to prove this result is inspired by the shape invariant for subsets of exact symplectic manifolds. As a corollary the author concludes the known result that the unit cotangent bundles of closed manifolds \(B\) with nowhere zero one-forms are strongly orderable for \(n>1\).