Lagrangean contact structures on projective cotangent bundles
Let \((M,D)\) be a contact manifold. A Lagrangian contact structure on \(M\) is a pair \((E, E')\) of subbundles of \(D\) such that for each point \(x \in M\) the fibres \(E_x\) and \(E_x'\) are transversal Lagrangian subspaces of \(D_x\) with respect to the natural conformal symplectic structure of \(D_x\). Let \(P(T^* N)\) be the projective cotangent bundle of a manifold \(N\) and \(D\) the canonical contact structure on \(P (T^* N)\); then, if a projective structure \(Q\) on \(N\) is given, \(P(T^* N)\) is endowed with a Lagrangian contact structure canonically associated to \(Q\). The following theorem is proved: the Lagrangian contact structure on \(P(T^* N)\) associated to a projective structure \(Q\) on \(N\) is flat if and only if \(Q\) is projectively flat. The proof makes use of the theory of \(G\)-structures associated to simple graded Lie algebras developed by \textit{N. Tanaka} [Hokkaido Math. J. 8, 23-84 (1979; Zbl 0409.17013)].
scientific article; zbMATH DE number 755427