Some aspects of differential geometry associated with hypoelliptic second order operators

Abstract

I investigate some aspects of the geometry of the ^characteristics of a class of hypoelliptic second order partial differential operators. The resulting geometry looks quite a bit like Riemannian geometry, although with interesting differences. 1. Introduction. In this paper, we investigate the relationship between certain problems in the calculus of variations and the geometry of the bicharacteristics of a second order hypoelliptic operator. Let M denote a connected C°° manifold of dimension m. Let Δ denote a second order hypoelliptic partial differential operator on M. We assume that the set of second order zeros of the principal symbol of Δ is a smooth submanifold of T*M. Since Δ is second order it is always possible, at least locally, to find a function V(x) and vector fields {gi}ni=0 such that Δ = ]C?=i Cfgf + go + V{x) for some constants cι = ±1. Hόrmander [19] has shown that a sufficient condition that Δ be hypoelliptic is that the sign of the c, doesn't change and that the evaluation map on vector fields is at each point x e M onto TXM when restricted to the Lie algebra of vector fields (over R) generated by the vector fields {gi}ni=0. In this paper I will be interested in hypoelliptic operators which satisfy the stronger condition that the Lie algebra generated by {g/} =1 is onto TM. Now, because Δ is of second order, Δ defines a quadratic form G* on T*M: if fx, f2 e C°°(M) and fx (JC) = f2(x) = 0, then