|Jun 15, 2015
In Riemannian geometry, the Liouville theorem says that the geodesic flow on the tangent bundle preserves the volume element of the tangent bundle of a Riemannian manifold. As a generalization of Riemannian structure, Eile Cartan introduced a concept of cone structures. A cone structure on a complex manifold is a closed nonsingular subvariety of the projetivized tangent bundle such that the natural projection is a submersion. In particular, the variety of minimal rational tangents on a uniruled projective manifold is an important example of cone structures. We say that a uniruled projective manifold have the Liouville property if every local vector field preserving the cone structure associated to the VMRT can be extended to a global holomorphic vector field on a Zariski open subset. I will give an introduction to this topic and show the Liouville property of horospherical varieties of Hermitian symmetric type.