Reeb Foliations on S and Contact 5-manifolds Violating the Thurston-bennequin Inequality

We obtain the following two results through foliation theoretic approaches including a review of Lawson’s construction of a codimension-one foliation on the 5-sphere: 1) The standard contact structure on the 5-sphere deforms to ‘Reeb foliations’. 2) We define a 5-dimensional Lutz tube which contains a plastikstufe. Inserting it into any contact 5-manifold, we obtain a contact structure which violates the Thurston-Bennequin inequality for a convex hypersurface with contact-type boundary.