Baire diagonal passage from finite regularity to smooth metrics

At a zero $(g,\gamma )$ of the universal section, write its surjective linearization as $L(\delta g,\xi )=A(\delta g)+D_{\gamma }\xi \mathrm{.}$ Then the tangent space to the universal zero set is $\ker L$, and the differential of its metric projection is $(\delta g,\xi )\mapsto \delta g$. If $D_{\gamma }$ is onto, this projection is onto by solving $D_{\gamma }\xi =-A(\delta g)$. Conversely, if the projection is onto, write any target vector as $A(\delta g)+D_{\gamma }\xi$ using surjectivity of $L$, choose $(\delta g,\eta )\in \ker L$, and subtract to obtain $D_{\gamma }(\xi -\eta )$ equal to that target. Hence a metric is a regular value of the projection exactly when every corresponding $D_{\gamma }$ is onto, which by [F4] is exactly the required stable--unstable transversality.

If $p=q$, the stable and unstable tangent spaces at $p$ are complementary eigenspaces of the hyperbolic gradient linearization; strict descent excludes any other intersection. If $p\mathrm{\ne }q$ and $f(p)\le f(q)$, strict descent makes the intersection empty. These cases hold for every metric. Hence suppose $f(p)>f(q)$.

Near each metric $g_{\ast }$ choose compact local unstable and stable disks at $p,q$, with parametrizations depending continuously in $C^1$ on $g$ in a $C^2$ neighbourhood $U$. Here is the parameter dependence needed: in coordinates near either zero fix its hyperbolic linearization $L$ at $g_{\ast }$ and write $X_g(x)=Lx+R_g(x)$. After a cutoff on a sufficiently small ball, $R_g$ has uniformly small Lipschitz constant. For prescribed small stable coordinate $z$, the integral equation $x(t)=e^{tL}z+{\int }_0^t{e}^{(t-s)L}{P}_s{R}_g(x(s))ds-{\int }_t^{\mathrm{\infty }}{e}^{(t-s)L}{P}_u{R}_g(x(s))ds$ is a uniform contraction on a small exponentially weighted space of paths on $[0,\mathrm{\infty })$. Its fixed point and its derivative with respect to $z$ depend continuously on $g$; the derivative solves a linear contraction equation. Evaluation at zero gives the stable disk. Reversing time gives the unstable disk. Restrict to smaller closed parameter balls so both disks extend beyond their boundaries. Finite-time flows also depend continuously in $C^1$ on $g$.

For integers $j,k\ge 0$, let $T_{jk}\subset U$ require transversality of the disk maps ${\mathrm{\Phi }}_j^g{D}_g^u(p)$ and ${\mathrm{\Phi }}_{-k}^g{D}_g^s(q)$ at every coincidence of points in their compact parameter domains. Tangent spaces here are those of the extended disks, including at boundary points. This is an open condition: if failing metrics converged in $C^2$ to a metric satisfying it, compactness would give convergent coincidence parameters, and the closed rank-deficiency condition would contradict transversality at their limit. Every point of a global stable or unstable manifold eventually flows into the interior of the corresponding local disk. Thus all $T_{jk}$ together are equivalent to the desired global transversality on $U$.

Each $T_{jk}\cap {\mathcal{G}}^{\mathrm{\infty }}$ is dense in $U\cap {\mathcal{G}}^{\mathrm{\infty }}$. Indeed, start at any smooth $g_0$ in a specified basic smooth neighbourhood controlling derivatives through order $r$. Choose finite $H\ge \max \{r,2\}$ above the Sard--Smale threshold. Fix $g_0$ on small disjoint critical neighbourhoods. Every trajectory between distinct critical points leaves their union, so metric variations outside them are the variations allowed in [F1]. By [F1], [F2], step 1.1 and Baire, there are arbitrarily $C^H$-close metrics in this affine Banach parameter space for which the pair is transverse. Choose one, $g_1$, still in $U$ and within the prescribed derivative bounds; it satisfies $T_{jk}$.

Approximate $g_1$ by a smooth symmetric tensor using convolution in a finite coordinate cover and a smooth partition of unity. This converges in $C^H$ because $H$ is finite and $M$ is compact. Sufficiently close approximants remain positive definite, remain in the prescribed neighbourhood, and still satisfy $T_{jk}$ by its $C^2$ openness. Only this compact-disk condition needs to survive smoothing. Since $g_0$ was arbitrary, this proves the claimed smooth density without fixing any common critical-neighbourhood data throughout $U$.

The smooth symmetric tensors on compact $M$ form a separable complete metrizable space under their countable derivative seminorms; positivity is an open condition. Hence ${\mathcal{G}}^{\mathrm{\infty }}$ is second countable and completely metrizable (an open subset admits an equivalent complete metric). Choose countably many neighbourhoods $U_i$ as above and closed sets $V_i\subset U_i\cap {\mathcal{G}}^{\mathrm{\infty }}$ whose interiors cover ${\mathcal{G}}^{\mathrm{\infty }}$. This is obtained by taking sufficiently small closed balls from a countable metric basis. For each $i,j,k$, set $O_{ijk}=({\mathcal{G}}^{\mathrm{\infty }}\setminus V_i)\cup (T_{i,jk}\cap {\mathcal{G}}^{\mathrm{\infty }})\mathrm{.}$ These sets are open and dense by steps 2.1 and 4.1.

By [F3] and dependent choice, ${\bigcap }_{i,j,k}{O}_{ijk}$ is dense and residual. Any metric in this intersection belongs to some $V_i$, hence satisfies every $T_{i,jk}$ and therefore the global pair transversality by step 2.1. The complement of the desired set is consequently contained in a countable union of closed nowhere dense sets, so the desired set itself is residual. Together with step 1.2 this covers all ordered pairs.