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 $f: X \to \mathbf R$ such that $df(v) > 0$. Let $\partial X$ be the boundary of $X$. We denote by $C^\infty(X)$ the algebra of smooth functions on $X$ and by $C^\infty(\partial X)$ the algebra of smooth functions on $\partial X$. With the help of $(v, f)$, we introduce two subalgebras $\mathcal A(v)$ and $\mathcal B(f)$ of $C^\infty(\partial X)$ and prove (under mild hypotheses) that $C^\infty(X) \approx \mathcal A(v) \hat\otimes \mathcal B(f)$, the topological tensor product. Thus the topological algebras $\mathcal A(v)$ and $\mathcal B(f)$, \emph{viewed as boundary data}, allow for a reconstruction of $C^\infty(X)$. As a result, $\mathcal A(v)$ and $\mathcal B(f)$ allow for the recovery of the smooth topological type of the bulk $X$.