The topology of a subspace of the Legendrian curves on a closed contact 3-manifold (Q2923145)

Let \(M\) be a \(3\)-dimensional smooth closed orientable contact manifold, and let \(\alpha\) be a contact \(1\)-form on it such that \(\omega =\alpha \wedge d\alpha\) is a volume form. Let \(\mathcal{L}_{\alpha}\) denote the subspace of the free loop space \(\varLambda (S^1,M)\) consisting of all Legendrian closed curves. It is known that the inclusion map \(j:\mathcal{L}_{\alpha}\to \varLambda (S^1,M)\) is an \(S^1\)-equivariant homotopy equivalence. In this paper the authors study the generalization of this result. For this purpose, they consider the condition that there exists a smooth vector field \( \nu \in \mathrm{Ker }(\alpha)\) such that the dual \(1\)-form \(\beta =d\alpha (\nu ,\cdot )\) is a contact form with the same orientation as \(\alpha\). In this situation, by renormalizing \(\nu\) suitably, we can also assume that \(\omega =\beta \wedge d\beta\). Moreover, they consider the condition that if starting from any \(x_0\) in \(M\), the rotation of \(\mathrm{Ker }(\alpha)\) along the \(\nu\)-orbit in a moving frame exceeds \(\pi\). Under these two conditions they study the subspace \(C_{\beta}\subset \mathcal{L}_{\beta}\) given by \(C_{\beta}=\{x\in \mathcal{L}_{\beta}:\alpha_{x}(\dot{x})=c\}\), where \(c\) is a constant that varies with the curve \(x\). Then they prove that the inclusion map \(i: C_{\beta}\to \varLambda (S^1,M)\) is an \(S^1\)-equivariant homotopy equivalence if the above two conditions are satisfied.