Legendrian Submanifold Path Geometry

Abstract

Let $Z\to Y^{2n+1}$ be the bundle of Legendrian $n$-planes over a contact manifold $Y$. We consider a foliation of $Z$ by canonical lifts of Legendrian submanifolds, called \emph{Legendrian submanifold path geometry}, whose flat model is \[ Sp(n+1, R) \to RP^{2n+1}. \] The equivalence problem provides an $sp(n+1,R)$ valued Cartan connection form that captures the geometry of such foliations. Two special cases are considered. The first case is characterized by having a well defined conformal class of symmetric $(n+1)$ differentials on the space of leaves of the foliation $X$. The $G$ structure induced on $X$ gives an example of a classical non-metric, irreducible holonomy $GL(n+1,R)$ with representation on $sy{m}^2(R^{n+1})$. In the second example, we consider a \emph{Legendrian} connection on the contact hyperplane vector bundle over $Y$ whose \emph{geodesic} Legendrian submanifolds give rise to a desired foliation on $Z$. There exists a unique \emph{normal symplectic} connection associated to a Legendrian connection analogous to the normal projective connection for a torsion free affine connection. For a nonflat example with symmetry, consider a hypersurface $M^n$ in the $(n+1)$ dimensional space form ${\bar{M}}_c^{n+1}$, $c=1,0,$ or -1, without any extrinsic symmetry. The images of $M$ under the motion by Iso(${\bar{M}}_c^{n+1}$), when lifted, generates a Legendrian submanifold path geometry on $Gr(n,{\bar{M}}_c^{n+1})$.