The Existence Theorem in Sub-Lorentzian Geometry (Joint Work with L.V. Lokutsievskiy)

A Lorentzian structure on a smooth manifold of dimension n is defined by a non-degenerate quadratic form with signature (1,n) smoothly depending on the point of the manifold. In each tangent space, this quadratic form defines a cone, half of which is called the future cone (closed convex cone), and the other half is the past. Admissible velocities are contained within the future cone, and their length is determined by the quadratic form.
Does a longest curve connecting given points exist? Usual reasoning does not apply here, as the set of admissible velocities is not compact, and the integrand function of the quality functional is concave. The answer to this question is given in a uniform way for Lorentzian and sub-Lorentzian geometry (and even more general situations). Conditions for the existence of the longest are expressed in terms of the causal structure (1-forms on the manifold, defining the future cones).
Some examples will be discussed.