The Lie algebra of a smooth manifold

Abstract

It is well known that certain topological spaces are determined by rings of continuous real functions defined over them [1; 2; 3],1 and for differentiable manifolds the functions may be differen tiable [4; 7]. In this note we prove that the Lie algebra of all tangent vector fields with compact supports on an infinitely differentiable manifold determines the manifold, and that two such manifolds with isomorphic Lie algebras are differentiably homeomorphic. This Lie algebra has been studied by H. Cartan [6], and for the case of analytic manifolds by Chevalley [5]. Let X be an infinitely differentiable (smooth) manifold and let R be the real numbers. Denote by D the algebra over R of all infinitely differentiable functions on X with the natural addition and multiplication, and by Do the subalgebra of D consisting of those functions with compact supports (that is, which vanish outside of compact sets). A tangent vector field L is a linear operator on D to D which is also an abstract derivative, that is, if a, bER and f, gED, then L(af +bg) =aL(f) +bL(g) and L(fg) =fL(g) +gL(f). The set et of all tangent vector fields, with the obvious addition and multiplication by real numbers, becomes a Lie algebra2 over R when the product of L1 and L2 is defined by