= Solution
Here “bumpy” means a <nowhere locally homogeneous metric>, not the different convention concerning nondegenerate closed <geodesics>. A smooth <Riemannian metric> has this property if every <local isometry> between connected open subsets is the identity on its domain; in particular, no two distinct open subsets are isometric. The <Sunada local isometry lemma> states that, on a compact smooth <manifold> without boundary of dimension at least two, such <Riemannian metrics> form a <residual set> and are dense in the $C^\infty$ topology. The dimension restriction matters: in dimension one, arclength coordinates always give nontrivial <local isometries>.
An element $j_x^kf$ of the <jet bundle of maps> $J^k(M,N)$ records the source point $x$, target value $f(x)$ and all derivatives through order $k$. If $\dim M=m$ and $\dim N=n$, there are $\binom{m+r-1}{r}$ derivative <multi-indices> of order $r$, and $n$ components for each one. Thus
$$
\boxed{\dim J^k(M,N)=m+n\binom{m+k}{k}.}
$$
For the projection $j_x^kf\mapsto(x,f(x))$ onto $M\times N$, the fiber has dimension $n(\binom{m+k}{k}-1)$. In source and target coordinates its Taylor coefficients identify it with
$$
\boxed{\bigoplus_{r=1}^k\operatorname{Sym}^r(T_x^*M)\otimes T_yN.}
$$
This is a coordinate-dependent identification for $k>1$, since a change of coordinates mixes different derivative orders. For $k=1$ the fiber is canonically $\operatorname{Hom}(T_xM,T_yN)$; for higher orders, truncation to the previous <jet bundle of maps> is an <affine bundle> modeled on $\operatorname{Sym}^k(T_x^*M)\otimes T_yN$. For $k=0$ the fiber is a point.
\b[The density assertion in the printed question requires distinct closed domains.] Literally, for $i=k$, the identity is an <isometry> for every <Riemannian metric>, so $\mathcal S_{ii}$ is the whole metric space and its complement is empty. Merely indexing the same closed ball twice has the same defect. The intended assertion is for distinct embedded closed balls $K_i=\overline U_i$ and $K_k=\overline U_k$, with an <isometry> meaning an onto <diffeomorphism> preserving the metric. The proof below establishes precisely this corrected assertion.
Distinct embedded closed balls are regular closed sets, so at least one of $\operatorname{int}K_i\setminus K_k$ and $\operatorname{int}K_k\setminus K_i$ contains a nonempty open set. Indeed, if both were empty, taking closures would give both inclusions and hence equality. Interchange the labels if necessary and choose a nonzero nonnegative <smooth bump function> $\eta$ supported in $\operatorname{int}K_i\setminus K_k$. For any given <Riemannian metric> $g$, make the <localized volume perturbation>
$$
g_t=e^{2t\eta}g.
$$
The <Riemannian volume form> satisfies $dV_{g_t}=e^{mt\eta}dV_g$. Therefore
$$
\operatorname{vol}_{g_t}(K_k)=\operatorname{vol}_g(K_k),\qquad
\frac d{dt}\operatorname{vol}_{g_t}(K_i)=m\int_{K_i}\eta e^{mt\eta}\,dV_g>0.
$$
If the original volumes agree, every sufficiently small $t>0$ makes them unequal. If they already disagree, continuity preserves that inequality for sufficiently small $t$. An onto <isometry> must preserve volume, so each of these perturbed <Riemannian metrics> belongs to $\mathcal C\mathcal S_{ik}$. Finally $g_t\to g$ in $C^\infty$: all derivatives of $e^{2t\eta}-1$ tend to zero uniformly on compact sets. This proves \b[density of $\mathcal C\mathcal S_{ik}$ for distinct closed balls], in fact in every positive dimension. The <localized volume perturbation> settles this particular density claim directly. It also does not imply the stronger <Sunada local isometry lemma> by itself, since a general <local isometry> need not send one preselected basis ball onto another.
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