Calibrated geodesic foliations of hyperbolic space

Let H be the hyperbolic space of dimension n + 1. A geodesic foliation of H is given by a smooth unit vector field on H all of whose integral curves are geodesics. Each geodesic foliation of H determines an n-dimensional submanifold of the 2n-dimensional manifold ℒ of all the oriented geodesics of H (up to orientation preserving reparametrizations). The space ℒ has a canonical split semi-Riemannian metric induced by the Killing form of the isometry group of H. Using a split special Lagrangian calibration, we study the volume maximization problem for a certain class of geometrically distinguished geodesic foliations, whose corresponding submanifolds of ℒ are space-like. 2010 Mathematics Subject Classification. Primary 53C38, 53C12, 53C22, 53C50. Key words and phrases. Split special Lagrangian calibration, geodesic foliation, hyperbolic space, space of geodesics.