Application of Fr olicher-Nijenhuis Theory in Geometric Characterization of Metric Legendre Foliations on Contact Manifolds

In the context of geometry and mathematical physics, the im- pression of Lagrangian foliations on symplectic manifolds is of speci c signi - cance. More recent is the study of the theory of Legendre foliation on contact manifolds which are geometrically reckoned as the analogues of Lagrangian foliations in the odd dimensional circumstances. In this paper, a compre- hensive analysis of the geometric organization of metric Legendre foliations on contact manifolds via the Fr olicher - Nijenhuis formalism is presented. For this purpose, the global expression of Helmholtz metrizability constraints expressed by an arbitrary semi-basic 1-form is applied in order to induce a metric structure which leads to construction of a Legendre foliation equipped with a bundle-like metric on an arbitrary contact manifold. Moreover, the local framework of metric Legendre foliations is exhaustively analyzed by ap- plying two signi cant local invariants existing on the tangent bundle of a Legendre foliation of the contact manifold ( M;ϖ ) ; One of them is a symmet- ric 2-form and the other one is a symmetric 3- form. Mainly, it is proved that under some particular circumstances the behaviour of the Legendre foliation on the contact manifold ( M;ϖ ) is locally the same as the foliation de ned by the determined distribution which is fundamentally perpendicular com- plement in TTM whose leaves are explicitly the c-indicatrix bundle de ned on M.