Morse--Smale pairs

Let $f:M\to \mathbb{R}$ be Morse and let $X$ be a complete downward gradient-like field. The pair $(f,X)$ is Morse--Smale if every unstable manifold $W^u(p)$ is transverse to every stable manifold $W^s(q)$, for critical points $p,q$.

The metric version also applies to any smooth Riemannian metric $g$ for which $X=-{\mathrm{grad}}_{g}f$ is complete. Here $W^u(p)$ and $W^s(p)$ mean the backward- and forward-limit manifolds of this negative-gradient flow, respectively, and $(f,g)$ is Morse--Smale when all these intersections are transverse. This metric version does not require $X$ to have the exact normalized Morse-coordinate form in the downward-gradient-like definition. Completeness is automatic on a closed manifold. In both versions an empty intersection is transverse vacuously.