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A finite isometry group decomposes $L^2(N)$ into its irreducible representation factors $V_{\rho }$ and multiplicity Hilbert spaces ${\mathcal{H}}_{\rho }$. The Laplacian commutes with the group, so it acts as the identity on each representation factor. The averaging projection gives $\dim V_{\rho }^{U_i}=|U_i{|}^{-1}{\sum }_{u\in U_i}{\chi }_{\rho }(u)$. For Gassmann equivalent subgroups these dimensions agree. Choosing unitary identifications of the invariant finite-dimensional factors therefore intertwines the quotient Laplacians, even when a noncompact covering manifold has no finite heat trace.

The lattice theta series of a full-rank Euclidean lattice records the multiset of squared lengths of lattice vectors. The Gaussian form shown here converges for $t>0$. For a flat torus ${\mathbb{R}}^d/\Lambda$, its heat trace is ${\Theta }_{{\Lambda }^{\ast }}(4\pi t)$, by the spectrum of a flat torus. Equal theta series do not in general imply orthogonal equivalence of lattices.

For a Laplace-type operator on a closed manifold, let $a_{t^0}$ be the constant coefficient of its full heat trace expansion. In the Mellin transform defining the spectral zeta function, the reduced trace has constant coefficient $a_{t^0}-\dim \ker L$. Its contribution is $(a_{t^0}-\dim \ker L)/s$, and $1/\Gamma (s)=s+O(s^2)$ proves the identity. For the scalar positive Laplace-Beltrami operator on a closed surface, $a_{t^0}=\int RdV/(24\pi )$ and $\dim \ker L$ is the number of connected components. In particular, a connected sphere has value $-2/3$ and a connected flat torus has value $-1$.