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 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~KS90 which dealt with the case of a constant cone in a vector space.