Legendre foliations on contact metric manifolds : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University

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Massey University
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This thesis develops the theory of Legendre foliations on contact manifolds by associating a contact metric structure with a contact manifold and investigating Legendre foliations on the resultant contact metric manifold. The contact metric structure introduces a metric for the Legendre foliation which enables us to study the curvature properties of a Legendre foliation, furthermore when this metric is bundle-like we have a semi-Riemannian foliation hence we can define a semi-Riemannian Legendre foliation and study its properties. We use the invariant Π as defined by Pang to define a family of contact metric structures for a non-degenerate Legendre foliation and from this family we pick out a unique contact metric structure the canonical contact metric structure. Furthermore a canonical contact metric structure is identified for a flat Legendre foliation and shown to be a Sasakian structure. Under some circumstances a Legendre foliation on a contact metric manifold has a second Legendre foliation, the conjugate Legendre foliation, associated with it. We investigate the conditions for the existence and the properties of the conjugate Legendre foliation. By using a definition similar to that of a Legendre foliation on a contact metric manifold we conclude this thesis by defining a complex Legendre foliation on a complex contact metric manifold and beginning an investigation of its properties.
Legendre foliations, Contact manifolds, Legendre polynomials