Gradient flow on a manifold

Summary

The gradient flow of a smooth function $f$ on a Riemannian manifold is the flow of the vector field $-\nabla f$. Some authors use $\nabla f$ instead. Along its flow lines the function decreases, strictly away from the critical points, which are exactly the rest points of the flow. It is the basic tool of Morse theory and of critical point theory: it deforms sublevel sets of $f$ and connects critical points.