I'm looking for a quick introduction to the idea of an angle function for a manifold. As a guess I think it's similar to a distance function, but maps tangent vectors rather than points to some value.
As Zhen says, a topological or smooth manifold is not equipped with a notion of angle. To describe an angle requires a conformal structure on a manifold, which can be induced by a Riemannian metric. A Riemannian metric equips tangent spaces with an inner product, so one can speak of the angle between two tangent vectors.