On smooth time functions

Source abstract

Abstract We 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 defined a field of tangent convex cones ( C x ) x ∈ M enjoying mild continuity properties. Under some conditions on its integral curves, we will construct a time function. Our approach is based on the definition of an intrinsic length for curves indicating how a curve is far from being an integral trajectory of C x . We find connections with topics pertaining to Hamilton–Jacobi equations, and make use of tools and results issued from weak KAM theory.