Legendrian submanifold path geometry (Q1566334)

\textit{S. S. Chern} [Sci. Rep. Nat. Tsinghua Univ., A 4, 97-111 (1940; JFM 66.0879.02); see Selected papers. Vol. I, New York, Springer (1978; Zbl 0403.01012), 76-82] dealt with the foliation on the bundle \(Z^4\to Y^3\), where \(Z^4\) is the bundle of Legendrian line elements over a contact manifold \(Y^3\), defined by canonical lifts of Legendre curves. In this paper a generalization to higher dimension is attempted by considering the canonical lifts of Legendrian submanifolds. This is called the Legendrian submanifold path geometry, and a path here is a Legendrian \(n\)-fold. The equivalence problem yields an \({\mathfrak s}{\mathfrak p }(n+1)\)-valued Cartan connection form. In the structure equations associated to this path geometry, to special cases are considered. The first is characterized by having a conformal class of symmetric \((n+1)\)-differentials on the space of leaves of the foliation, while the second is the direct generalization of the normal projective connection to the path geometry. It is noted that all the arguments made above remain valid when one replaces real and smooth by complex and holomorphic.