In S. Smale's, “On gradient dynamical systems,” Ann. of Math. (2), vol. 74, no. 1, pp. 199–206, 1961, four conditions on a vector field $X$ on a compact manifold $M$ are given that are sufficient for the existence of a smooth function $f$ and a Riemannian metric $g$ such that $X=\text{grad}_gf$.

I'm confused about the relationship of conditions (2) and (3). Let $\beta_1,...,\beta_m$ denote the zeros of $X$. Smale writes:

(2) If $x\in\partial M$, $X$ is transversal to $\partial M$. Hence $X$ is not zero on $\partial M$.
(3) If $x\in M$ let $\phi_t(x)$ denote the orbit of $X$ (solution curve) satisfying $\phi_0(x)=x$. Then for each $x\in M$, the limit set of $\phi_t(x)$ as $t\to\pm\infty$ is contained in the union of the $\beta_i$.

Now, if I interpret the orbits to be stopped at the boundary, it seems to me that if $x\in M$ is a point close to $\partial M$, either $\lim_{t\to\infty}\phi_t(x)$ or $\lim_{t\to-\infty}\phi_t(x)$ must be in $\partial M$ (depending on whether $X$ points in- or outwards). Therefore there has to be a $\beta_i$ on $\partial M$. However, this contradicts condition (2).

Therefore it must be the case that my interpretation of the orbits is incorrect. An interpretation that works is if we impose the restriction on the limits only when the orbit can be extended to either a half-line or the entire real line. This makes sense also in view of the fact that $X$ is not assumed to be complete.

Could someone clear this up for me? Is my second interpretation correct?


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