= Isospectral stabilization by a small flat torus
{title2=$M_i\times(\mathbb R^k/\varepsilon\mathbb Z^k)$}
Two <isospectral manifolds> remain isospectral after taking a <Riemannian product> with the same <closed> factor: <eigenvalues> add, with <multiplicities> given by the <product Laplacian eigenspace decomposition>. For nonisometric <closed> connected factors $M_1,M_2$, choose $\varepsilon<\delta<\min_i\operatorname{inj}(M_i)$. Every <closed> product <geodesic> of length less than $\delta$ has constant projection to $M_i$, since a nonconstant <closed geodesic> shorter than the <injectivity radius> is impossible. The short coordinate <geodesics> of the small <flat torus> span exactly its tangent factor at every point. Any product <Riemannian isometry> must preserve that intrinsically defined <smooth distribution> and its <orthogonal complement>, hence restrict to an <isometry> of the original factors. Thus these products are still nonisometric. Taking $M_i$ to be four-dimensional <tori> proves the higher-dimensional stabilization statement without assuming a cancellation theorem for arbitrary products.
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