The Hypo-elliptic Differential Forms On Smooth Manifolds

The hypoelliptic differential operators are a class of important partial differential operators, which have been widely studied.The Laplace operator and two-order constant coefficient elliptic operators are all hypoellipticity, The famous Hormander's sum of square differential operator is hypoelliptic if it satisfies the Hormander condition of rank,which are two orderdifferential operators . Vector fields on smooth manifolds can be viewed as differential operators acting on function space, their hypoellipticity have been studied from 1970's. There are many results concerning it. As the dual of vector fields, differential forms are also natural differential operators act on spaces of forms. We can consider their hypoellipticity too. A.Meziani has found a sufficient condition which ensures hypoellipticity of the closed forms on closed manifold. The contact forms on the sphere S~3 are hypoelliptic, but they aren't closed forms. Which states that there exists non-closed hypoelliptic forms on smooth manifolds which satisfy some condition. Here we make use of Ergodic theory and the solvability of partial differential equations to discuss hypoellipticity of differential forms. The conclusions which we have obtained show that the topological structure of manifolds has an obstruct to the existence of hypoelliptic differential forms.This paper is divided into six chapters as follows. Chapter one mainly introduces the background about hypoelliptic question and the conclusions we have obtained; Chapter two gives some relative basic knowledge and some examples about hypoelliptic differential forms; Chapter three includes some basic results and their proof; Chapter four shows a sufficient and necessary condition that the smooth form (?) = dx + λ(x, y)dy is hypoellipticity on the torus T~2 ; Chapter five gives the sufficient and necessary condition that smooth (n-1) forms are hypoellipticity on the torus T~n ; Chapter six discusses the connection about the hypoelliptic form with the first Betti number of manifold.