On smooth time functions

International audienceWe are concerned with the existence of smooth time functions on connected time{oriented Lorentzian manifolds. The problem is tackled in a more general abstract setting, namely in a manifold M where is just de ned a eld of tangent convex cones (Cx)x2M enjoying mild continuity properties. Under some conditions on its integral curves, we will construct a time function. Our approach is based on the de nition of an intrinsic length for curves indicating how a curve is far from being an integral trajectory of Cx. We nd connections with topics pertaining to Hamilton{Jacobi equations, and make use of tools and results issued from weak KAM theory