Riemannian product

For Riemannian manifolds $(M,g)$ and $(N,h)$, the product metric on $M\times N$ is $g\oplus h$. Its Levi-Civita connection and curvature split by factors, so its Ricci and scalar curvatures are the corresponding sums.

This equips a product manifold with the direct-sum Riemannian metric.

For closed manifolds with the product Riemannian metric, the positive Laplace-Beltrami operator is the sum of the two factor operators. Products of their orthonormal eigenbases form a complete basis, and eigenvalues add. Hence the product eigenspace at $\lambda$ is the displayed finite orthogonal sum of tensor products. The discrete assertion requires a discrete spectral setting.

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 $\epsilon <\delta <{\min }_i\mathrm{inj}(M_i)$. 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.