= Solution
Here a <nowhere locally homogeneous metric> means a smooth <Riemannian metric> with no nonidentity <local isometry>: an <isometry> between open subsets must fix every point of its domain. This is the strong local-rigidity meaning of “bumpy” used in this question, not the different convention about nondegenerate closed <geodesics>. <Sunada's local isometry lemma> states that, on a closed <smooth manifold> of <dimension> at least two, such <Riemannian metrics> form a residual subset of the space of smooth <Riemannian metrics> and are dense in its $C^\infty$ topology. In <dimension> one this conclusion is false: arclength coordinates give local translations for every <Riemannian metric>.
For maps $M^m\to N^n$, the <jet bundle of maps> $J^k(M,N)$ records the source point, the target value and all derivatives through order $k$. There are $\binom{m+k}{k}$ monomials of total degree at most $k$ in $m$ variables. Hence
$$
\boxed{\dim J^k(M,N)=m+n\binom{m+k}{k}.}
$$
Over $(x,y)\in M\times N$, the value is already fixed, so the fibre has <dimension>
$$
\boxed{n\left(\binom{m+k}{k}-1\right).}
$$
In local coordinates it is the space of <Taylor series> coefficients of orders one through $k$, represented by a <direct sum> of <symmetric powers>, $\bigoplus_{r=1}^k\operatorname{Sym}^r(T_x^*M)\otimes T_yN$. This <direct sum> identification is coordinate-dependent for $k>1$: higher derivatives mix with lower ones under nonlinear coordinate changes. Intrinsically, the truncation to order $k-1$ is an <affine bundle> modeled on $\operatorname{Sym}^k(T_x^*M)\otimes T_yN$. For $k=1$ the fibre is canonically $\operatorname{Hom}(T_xM,T_yN)$.
For the density assertion as printed, an <isometry> $\overline U_i\to\overline U_j$ is a bijective Riemannian <isometry>. A <localized volume perturbation> suffices; it is not necessary to prove the stronger generic local-rigidity lemma afresh. We take the coordinate-ball basis without repetitions, and each $U_i$ as the interior of its smooth closed-ball closure. Distinct such domains have a nonempty open difference in at least one direction. Indeed, if both were contained in the other's closure, their regular-open property $U=\operatorname{int}\overline U$ would make them equal.
Fix any <Riemannian metric> $g$. If the two domains have unequal volumes, they cannot be isometric, so $g$ already lies in the desired complement. Otherwise, after interchanging the two indices if necessary, choose a nonnegative nonzero <smooth bump function> $\psi$ supported in $U_i\setminus\overline U_j$, and set
$$
g_\epsilon=e^{2\epsilon\psi}g.
$$
These are positive-definite smooth <Riemannian metrics> tending to $g$ in the $C^\infty$ topology as $\epsilon\to0$. Their <Riemannian volume forms> satisfy $dV_{g_\epsilon}=e^{m\epsilon\psi}dV_g$. The <Riemannian metric> and volume on $U_j$ are unchanged, whereas
$$
\left.\frac d{d\epsilon}\operatorname{vol}_{g_\epsilon}(U_i)\right|_{\epsilon=0}
=m\int_{U_i}\psi\,dV_g>0.
$$
For every sufficiently small $\epsilon>0$, the volumes differ. Thus no <isometry> of the two closed domains exists, proving
$$
\boxed{\mathcal C\mathcal S_{ij}\text{ is dense in the space of smooth metrics}.}
$$
The volume difference is continuous in the smooth topology, so the unequal-volume <Riemannian metrics> even form an open dense subset of this complement. The regular-open assumption is also needed to justify this literal closed-domain assertion. As written, requiring only that each closure be a closed ball permits two distinct basis members with the same closure. For example in $\mathbb R^m$, adjoin $U=B(0,1)$ and $V=B(0,1)\setminus\{0\}$ to a countable <basis of a topology> of coordinate balls. Both are open and both closures are the same closed ball. The identity is an <isometry> of those closures for every <Riemannian metric>, so the indicated complement is empty. Thus the conclusion needs distinct closures, as in the usual basis of genuine coordinate balls used above. The no-repetitions convention is necessary as well: if the indexed basis repeats a domain, its identity map is always an <isometry> and the asserted complement for those two indices is empty. This volume argument concerns <isometries> onto the specified whole domains; the stronger assertion excluding arbitrary <local isometries> is the separate Sunada lemma stated above.
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