  Steady states are invaluable in the study of dynamical systems.
High-dimensional dynamical systems, due to a separation of time-scales, often
evolve towards a lower dimensional manifold $M$. We introduce an approach to
locate saddle points (and other fixed points) that utilizes gradient extremals
on such a priori unknown (Riemannian) manifolds, defined by adaptively sampled
point clouds, with local coordinates discovered on-the-fly through manifold
learning. The technique, which efficiently biases the dynamical system along a
curve (as opposed to exhaustively exploring the state space), requires
knowledge of a single minimum and the ability to sample around an arbitrary
point. We demonstrate the effectiveness of the technique on the M\"uller-Brown
potential mapped onto an unknown surface (namely, a sphere). Previous work
employed a similar algorithmic framework to find saddle points using Newton
trajectories and gentlest ascent dynamics; we therefore also offer a brief
comparison with these methods.
