  We study the phase space of a particle moving in the gravitational field of a
rotating black hole described by the Kerr metric from a geometrical
perspective. In particular, we show the construction of a multidimensional
generalization of the unstable periodic orbits, known as Normally Hyperbolic
Invariant Manifold, and its stable and unstable invariant manifolds that direct
the dynamics in the phase space. Those stable and unstable invariant manifolds
divide the phase space and are robust under perturbations. To visualize the
multidimensional invariant sets under the flow in the phase space, we use a
method based on the arclength of the trajectories in phase space known as
Lagrangian descriptors in the literature.
