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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 $j_x^{k}f$ 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 $\left(\genfrac{}{}{0ex}{}{m+r-1}{r}\right)$ derivative multi-indices of order $r$, and $n$ components for each one. ThusFor the projection $j_x^{k}f\mapsto (x,f(x))$ onto $M\times N$, the fiber has dimension $n((\genfrac{}{}{0ex}{}{m+k}{k})-1)$. In source and target coordinates its Taylor coefficients identify it withThis is a coordinate-dependent identification for $k>1$, since a change of coordinates mixes different derivative orders. For $k=1$ the fiber is canonically $\mathrm{Hom}(T_{x}M,T_{y}N)$; for higher orders, truncation to the previous jet bundle of maps is an affine bundle modeled on ${\mathrm{Sym}}^k(T_x^{\ast }M)\otimes T_{y}N$. For $k=0$ the fiber is a point.

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 $\mathrm{int}{K}_i\setminus K_k$ and $\mathrm{int}{K}_k\setminus K_i$ 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 $\mathrm{int}{K}_i\setminus K_k$. For any given Riemannian metric $g$, make the localized volume perturbationThe Riemannian volume form satisfies $d{V}_{g_t}=e^{mt\eta }d{V}_g$. ThereforeIf 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 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.

In this context a nowhere locally homogeneous metric, also called a bumpy metric, is a smooth Riemannian metric for which no two distinct nonempty open subsets are isometric with their induced metrics. Equivalently every local isometry between open subsets is the identity wherever defined: a nonidentity local isometry sends some point $x$ to a distinct point, and restriction to sufficiently small disjoint neighborhoods would violate the first formulation. This is the local-isometry meaning of the terminology here; degeneracy of periodic geodesics is a different use of the word bumpy.

Sunada's local isometry lemma states that on a compact smooth manifold without boundary of dimension $m\ge 2$ the nowhere locally homogeneous metrics contain a residual set in the space of smooth Riemannian metrics with its $C^{\infty }$ topology. In particular they are dense, by the Baire category theorem. A residual set is a countable intersection of open dense sets. The dimension assumption matters: every one-dimensional Riemannian metric is locally $d{s}^2$ in arclength coordinates and has local translations.

The jet bundle of maps $J^k(M,N)$ consists of equivalence classes $j_x^{k}f$ of smooth maps near $x$, where two maps agree to order $k$ at $x$ in coordinate charts. Its projection to $M\times N$ sends $j_x^{k}f$ to $(x,f(x))$. For a multi-index $\alpha$ in $m$ variables there are $\left(\genfrac{}{}{0ex}{}{m+r-1}{r}\right)$ derivatives of order $r$. Thus a coordinate chart consists of the source point, the target point, and $n$ coefficients for every derivative order $1$ through $k$. Summing the counts givesThis includes $J^0(M,N)=M\times N$. The fibre over $(x,y)$ is the space of truncated Taylor maps with fixed constant term $y$, locally modeled onFor $k=1$ its identification with $\mathrm{Hom}(T_{x}M,T_{y}N)$ is canonical. For higher $k$, changes of target coordinates mix derivatives of different orders, so this is a coordinate or connection-dependent description, not a canonical vector-bundle identification. The truncation $J^k\to J^{k-1}$ is an affine bundle modeled on the pullback of ${\mathrm{Sym}}^k(T^{\ast }M)\otimes TN$ over $M\times N$.

For the density assertion put $A={\overline{U}}_i$ and $B={\overline{U}}_j$. Their closures must be distinct. If $A=B$, the identity is an isometry for every Riemannian metric, and the requested complement is empty. Under the intended distinct-domain assumption, the smooth closed-ball hypothesis makes $A$ and $B$ regular closed domains, equal to the closures of their interiors. After interchanging them if necessary, there is a nonempty open set $W$ with compact closure in $\mathrm{int}(A)\setminus B$. Otherwise each interior would be contained in the other closure, forcing $A=B$.

Take any smooth Riemannian metric $g$. If ${\mathrm{vol}}_g(A)\ne {\mathrm{vol}}_g(B)$, it already lies outside ${\mathcal{S}}_{ij}$, because an isometry preserves the Riemannian volume form. If the volumes agree, choose a nonzero nonnegative smooth bump function $\eta$ supported in $W$ and setThese are positive definite Riemannian metrics, they agree with $g$ on $B$, and $g_t\to g$ in $C^{\infty }$ as $t\downarrow 0$, since every derivative of $g_t-g$ is $t$ times a fixed compactly supported smooth tensor. Their Riemannian volume forms satisfyHence ${\mathrm{vol}}_{g_t}(B)={\mathrm{vol}}_g(B)$ whereas ${\mathrm{vol}}_{g_t}(A)>{\mathrm{vol}}_g(A)$ for every $t>0$. In particularThe two domains cannot be isometric for $g_t$. Every $C^{\infty }$ neighborhood of $g$ therefore meets the complement of ${\mathcal{S}}_{ij}$, provingThis localized volume perturbation proves the requested density even when the two domains overlap, and works in every positive dimension. It does not by itself prove the stronger residual local-isometry statement: fixed isometric closures and arbitrary isometric open subsets are different conditions.

On a compact smooth manifold without boundary of dimension at least two, a residual set of smooth Riemannian metrics consists of nowhere locally homogeneous metrics. Thus such metrics are dense in the smooth topology by the Baire category theorem. In dimension one, arclength coordinates always provide nontrivial local isometries.