Topologies and sheaves on causal manifolds

A causal manifold $(M,γ)$ is a manifold $M$ endowed with a closed proper cone $γ$ in the tangent bundle $TM$ such that the projection $TM\to M$ is surjective when restricted to the interior of $γ$. Let $λ$ be the antipodal of the polar cone of $γ$. An open set $U$ of $M$ is called $γ$-open if its Whitney normal cone contains the interior of $γ$. Similarly, $U$ is called $λ$-open if the micro-support of the constant sheaf on $U$ is contained in $λ$. We begin by proving that the two notions coincide. Next, we prove that if $(M,γ)$ admits a ``future time function'' the functor of direct images establishes an equivalence of triangulated categories between the derived category of sheaves on $M$ micro-supported by $λ$ and the derived category of sheaves on the manifold $M$ endowed with the $γ$-topology. This generalizes a result of~\cite{KS90} which dealt with the case of a constant cone in a vector space.