Topologies and Sheaves on Causal Manifolds

Abstract A causal manifold $(M,\gamma )$ is a manifold $M$ endowed with a closed convex proper cone $\gamma $ in the tangent bundle $TM$ satisfying suitable properties. Let $\lambda ={\gamma }^{\circ a}\subset T^{*}M$ be the antipodal of the polar cone of $\gamma $. An open set $U$ of $M$ is called $\gamma $-open if its Whitney normal cone contains $\textrm{Int}(\gamma )$. Similarly, $U$ is called $\lambda $-open if the micro-support of the constant sheaf on $U$ is contained in $\lambda $. We begin by proving that the two notions coincide. Next, we prove that if $(M,\gamma )$ admits a “future time function” (in particular if it is globally hyperbolic) the functor of direct images establishes an equivalence of triangulated categories between the derived category of sheaves on $M$ micro-supported by $\lambda $ and the derived category of sheaves on $M_\gamma $, the manifold $M$ endowed with the $\gamma $-topology. This generalizes a result of [5], which treated the case of a constant cone in a vector space.