The canonical contact structure on the space of oriented null geodesics of pseudospheres and products (Q2858023)

Let \(N\) be a complete pseudo-Riemannian manifold. If \(\gamma_{u}\) denotes the geodesic in \(N\) with initial velocity \(u\), then geodesics \(\gamma_{u}\) and \(\gamma_{v}\) are said to be equivalent if \(v = \lambda \gamma_{u}'(t)\) for some \(\lambda > 0\) and some \(t \in \mathbb{R}\). For \(r \in \mathbb{R}\) let \(T^{r}(N)\) denote the vectors in \(TN\) with length \(r\). Let \(\mathcal{L}^{0}(N)\) denote the space of equivalence classes in \(T^{0}(N)\) of null geodesics. If \(\mathcal{L}^{0}(N)\) is a manifold, then the canonical projection \(\Pi : T^{0}(N) \rightarrow \mathcal{L}^{0}(N)\) is a smooth submersion. In this case \textit{B. Khesin} and \textit{S. Tabachnikov} [Adv. Math. 221, No. 4, 1364--1396 (2009; Zbl 1173.37037)] have used the canonical 1-form on \(TN\) to construct a canonical contact structure on \(\mathcal{L}^{0}(N)\). \newline \noindent Given Riemannian manifolds \(M\), \(N\) one obtains a pseudo-Riemannian structure \(M_{+} \times N_{-}\) on \(M \times N\) by defining \(|(x,y)|^{2} = |x|^{2} - |y|^{2}\) for every vector \((x,y) \in T_{(u,v)} M \times N\). Let \(S^{k,m}\) denote the unit vectors in \(\mathbb{R}^{k+1}_{+} \times \mathbb{R}^{m}_{-}\). Let \(S^{j}\) denote the unit sphere in \(\mathbb{R}^{j+1}\). \newline \noindent Among various results the authors prove the following :NEWLINENEWLINE {Theorem 1}: \(\mathcal{L}^{0}(S^{k,m})\) is a manifold contactomorphic to \(T^{1}(S^{k}_{+} \times S^{m-1}_{-})\).NEWLINENEWLINE{Theorem 2}: Let \(M\) and \(N\) be complete Riemannian manifolds. Assume that the geodesic flow in \(TM\) is free and proper, and let \(\mathcal{L}(M)\) be the manifold of oriented geodesics of \(M\). Then 1) \(\mathcal{L}^{0}(M_{+} \times N_{-})\) is a manifold; 2) If there exists a smooth section \(S : \mathcal{L}(M) \rightarrow T^{1}M\), then S induces a contactomorphism between \(\mathcal{L}(M) \times T^{1}N\) and \(\mathcal{L}^{0}(M_{+} \times N_{-})\). The authors also obtain some results on the null-billiards of complete pseudo-Riemannian manifolds.