Algebras of Smooth Functions and Holography of Traversing Flows

Abstract

Let X be a smooth compact manifold and v a vector field on X which admits a smooth function f : X ! R such that df(v) > 0. Let @X be the boundary of X. We denote by C1(X) the algebra of smooth functions on X and by C1(@X) the algebra of smooth functions on @X. With the help of (v; f), we introduce two subalgebras A(v) and B(f) of C1(@X) and prove (under mild hypotheses) that C1(X) _ A(v) ^B(f), the topological tensor product. Thus the topological algebras A(v) and B(f), viewed as boundary data, allow for a reconstruction of C1(X). As a result, A(v) and B(f) allow for the recovery of the smooth topological type of the bulk X.