Stable Unstable Manifolds and Morse Smale Transversality — Examples

The complement of $\{0,\pi \}$ has two connected arcs. On each, $\sin \theta$ has fixed sign and every orbit has backward limit $0$ and forward limit $\pi$, so each arc is one time-translation orbit.

Here $W^u(p)=S^1\setminus \{q\}$ and $W^s(q)=S^1\setminus \{p\}$. Along their two open-arc intersection both tangent spaces equal $T{S}^1$, so their sum is $T{S}^1$. The reversed distinct pair has empty intersection, and at each equal critical-point pair one of the stable or unstable tangent spaces is $T{S}^1$. Thus all stable--unstable intersections are transverse and the specified standard-metric pair is Morse--Smale by Morse--Smale pairs.

Hence $\mathcal{M}(p,q)$ has exactly two points. This agrees with [F1] and directly displays the quotient by translation.

The negative-gradient equations are $\dot{u}=-\cos u\mathrm{/}(SA)$ and $\dot{v}=(\sin v\sin u-\epsilon \cos v)\mathrm{/}S$. The field is complete on this compact torus. Critical points require $u=\pi \mathrm{/}2$ or $3\pi \mathrm{/}2$. For $u=\pi \mathrm{/}2$ they have $v=\varphi ,\pi +\varphi$; for $u=3\pi \mathrm{/}2$ they have $v=2\pi -\varphi ,\pi -\varphi$. The mixed Hessian entry is zero, the $uu$ entry is $-A\sin u\mathrm{/}S$, and the $vv$ entry is $-(\cos v\sin u+\epsilon \sin v)\mathrm{/}S$, which equals $\pm 1$ at these points. Thus all four are nondegenerate: a maximum, an upper saddle $a=(\pi \mathrm{/}2,\pi +\varphi )$, a minimum, and a lower saddle $b=(3\pi \mathrm{/}2,\pi -\varphi )$, respectively. Their saddle values are $(2-S)\mathrm{/}S$ and $-(2-S)\mathrm{/}S$.

The closed strip $\pi \le v\le 2\pi$ is forward invariant: on its lower boundary $\dot{v}=\epsilon \mathrm{/}S>0$, and on its upper boundary $\dot{v}=-\epsilon \mathrm{/}S<0$. Since $a$ lies in its interior, every orbit with backward limit $a$ stays in this strip, and cannot have forward limit $b$, which is outside it. The reverse connection is excluded by strict decrease of $f$. Thus there are no connections between the distinct saddles.

On a surface all other nonempty intersections are automatically transverse: an unstable manifold of a maximum or stable manifold of a minimum is open; the remaining extremal stable/unstable manifolds are singletons and meet only their own complementary open manifold. At each saddle its stable and unstable tangent lines at that saddle are the complementary negative-gradient eigenspaces. A nonconstant orbit cannot have identical endpoints because $f$ strictly decreases. Together with step 2.1 these observations cover every pair and prove $(f,g)$ Morse--Smale.

A connecting trajectory has $df(X)<0$, so its regular-level slice is a transverse hypersurface in the parametrized intersection. By [F2] and [F3] the space modulo time has dimension $\lambda (p)-\lambda (q)-1$. This is zero for index drop one and one for the maximum-to-minimum pair. In particular, the latter slice is not a finite set: the open basins of the maximum and minimum overlap, since their complements are the finitely many saddle separatrices and critical points.

Put $A=R+r\cos v>0$. Differentiating the embedding gives $g=A^{2}d{u}^2+r^{2}d{v}^2$, and hence $\dot{u}=-\cos u\mathrm{/}A$, $\dot{v}=\sin v\sin u\mathrm{/}r$. This smooth field is complete because the torus is compact.

The critical equations are $\cos u=0$ and $\sin v=0$, giving exactly four points. At each, the Hessian in $(u,v)$ is diagonal with entries $-A\sin u$ and $-r\cos v\sin u$. Both entries are nonzero. Thus $(\pi \mathrm{/}2,0)$ is a maximum, $(3\pi \mathrm{/}2,0)$ a minimum, and $a=(\pi \mathrm{/}2,\pi )$ and $b=(3\pi \mathrm{/}2,\pi )$ are saddles; in particular $f$ is Morse everywhere.

The circle $v=\pi$ is invariant. On its interval $\pi \mathrm{/}2<u<3\pi \mathrm{/}2$, $\dot{u}=-\cos u\mathrm{/}(R-r)>0$, so every point has backward limit $a$ and forward limit $b$. Locally at $a$, writing $w=v-\pi$ gives $\dot{w}=-\sin w\sin u\mathrm{/}r$; for $w\mathrm{\ne }0$ small this forces $\mathrm{\mid }w\mathrm{\mid }$ to increase backwards while $u$ stays near $\pi \mathrm{/}2$. Thus a backward-converging trajectory must have $w=0$. At $b$ the same equation forces nonzero $\mathrm{\mid }w\mathrm{\mid }$ to increase forwards, so a forward-converging trajectory must also have $w=0$. Uniqueness and flow transport show that the indicated unstable and stable branches are precisely arcs of this circle.

Consequently, at every point of this open interval, $T{W}^u(a)=T{W}^s(b)=\mathbb{R}{\mathrm{\partial }}_u$. Their tangent sum has dimension one, whereas the torus has dimension two. The metric pair $(f,g)$ therefore fails the Morse--Smale condition.

Near $p_n$ and $q_n$ the displayed local fields are the required Morse normal forms; off them every patched summand has $d{f}_n(X_n)<0$. The coefficients are bounded on fixed supports and the outside coefficient has absolute value at most $1\mathrm{/}2$, so every $X_n$ is complete.

On each line the interval $(-1,1)$ is one flow orbit from $p_n$ to $q_n$. Thus $\mathcal{M}(p_n,q_n)$ is a singleton and hence is discrete, without any appeal to Morse--Smale transversality.

Therefore ${\mathcal{M}}_1(X):={\coprod }_{n\ge 1}\mathcal{M}(p_n,q_n)$ is an infinite discrete set. It refutes finiteness of the global index-one locus without a compactness condition, not finiteness for one fixed endpoint pair.

The two arcs from $0$ to $\pi$ cross $f^{-1}(0)$ respectively at $\pi \mathrm{/}2$ and $3\pi \mathrm{/}2$, and strict descent prevents a second crossing.

By [F1], these two slice points are exactly the two unparametrized trajectory classes.