application of frolicher-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 specic signi-cance. more recent is the study of the theory of legendre foliation on contactmanifolds which are geometrically reckoned as the analogues of lagrangianfoliations in the odd dimensional circumstances. in this paper, a compre-hensive analysis of the geometric organization of metric legendre foliationson contact manifolds via the frolicher - nijenhuis formalism is presented.for this purpose, the global expression of helmholtz metrizability constraintsexpressed by an arbitrary semi-basic 1-form is applied in order to induce ametric structure which leads to construction of a legendre foliation equippedwith a bundle-like metric on an arbitrary contact manifold. moreover, thelocal framework of metric legendre foliations is exhaustively analyzed by ap-plying two signicant local invariants existing on the tangent bundle of alegendre 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 thatunder some particular circumstances the behaviour of the legendre foliationon the contact manifold(m;ϖ)is locally the same as the foliation denedby the determined distribution which is fundamentally perpendicular com-plement in ttm whose leaves are explicitly the c-indicatrix bundle denedon m.