Gradientlike

Gradientlike refers to a smooth vector field that is compatible with a smooth function in the sense used in Morse theory and differential topology. Let M be a smooth manifold and f: M → R a Morse function (all critical points are nondegenerate). A vector field X on M is gradientlike for f if X has the following properties: X vanishes precisely at the critical points of f, and at every point p where df(p) ≠ 0 one has df(p)(X(p)) > 0, i.e., f increases along the flow of X away from critical points. Moreover, near each critical point p, X can be chosen to resemble the standard linear model given by the quadratic form that describes the local behavior of f, so that the local dynamics reflect the Morse index of p.

Relation to gradients and metrics. If g is a Riemannian metric on M, the gradient vector field grad_g f is gradientlike for f. Conversely, given a gradientlike X for f, one can often adjust the metric so that X becomes the genuine gradient of f with respect to that metric on a neighborhood of each critical point; outside those neighborhoods X f still increases along trajectories. This flexibility makes gradientlike fields a convenient tool in Morse theory, where one studies the topology of M via the flow lines of X.

Applications. Gradientlike vector fields are used to construct Morse complexes: critical points serve as generators and flow lines between critical points of successive indices give boundary maps. The theory often requires a gradientlike field to satisfy transversality conditions (Morse-Smale condition), ensuring stable and unstable manifolds intersect cleanly. They also underpin handle decompositions and proofs of Morse inequalities.

Example. On R^n with f(x) = ||x||^2 and the standard metric, the gradient is ∇f(x) = 2x, which is gradientlike for f with a single critical point at 0. See also Morse theory, gradient flow, Morse-Smale systems.