Gradient flow on a manifold

Remarks

Along a flow line, $\frac{d}{dt} f(\varphi_t(x)) = -\abs{\nabla f}^2 \le 0$, with equality exactly at critical points of $f$. So $f$ is a Lyapunov function for the flow, and the rest points of the flow are the critical points of $f$. If $M$ is compact, the flow is defined for all $t \in \R$.

Example: on $\R^2$ with the Euclidean metric, let $f(x, y) = x^2 - y^2$. Then $-\nabla f = (-2x, 2y)$ and $\varphi_t(x, y) = (e^{-2t} x, e^{2t} y)$. The origin is the only rest point, and the flow has the familiar saddle picture.

Non-example: on $\R^2$, the rotation field $X(x, y) = (-y, x)$ is not $-\nabla f$ for any smooth $f$. Its flow has closed orbits, but $f$ would strictly decrease along a nonconstant flow line of a gradient flow, so it could not return to its starting point.

The flow depends on the metric $g$, not only on $f$. A vector field $X$ with $df(X) < 0$ away from the critical points, and of a standard form near them, is called gradient-like for $f$. Such fields serve the same purposes in Morse theory. For a Morse function and a generic metric, the gradient flow is a Morse–Smale system.