2000/11/18 by Sung Ho Wang, Wang, Sung Ho
Mathematics · #53XX #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG #msc:53XX
paper · pdf · doi:10.48550/arxiv.math/0011135
arxiv created 2001/04/20 · arxiv updated 2009/11/30
Let Z → Y2n+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 Legendrian submanifold path geometry, whose flat model is Sp(n+1, R) → RP2n+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 sym2(Rn+1). In the second example, we consider a Legendrian connection on the contact hyperplane vector bundle over Y whose geodesic Legendrian submanifolds give rise to a desired foliation on Z. There exists a unique 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 Mn in the (n+1) dimensional space form Mcn+1, c=1, 0, or -1, without any extrinsic symmetry. The images of M under the motion by Iso(Mcn+1), when lifted, generates a Legendrian submanifold path geometry on Gr(n, Mcn+1).