= Spectral determination of hyperbolic curvature on a surface
{title2=$\int(K+1)^2\,dA=0$}
On a closed two-dimensional <Riemannian manifold>, the first three integrated scalar <heat invariants> are $A$, $\tfrac13\int K\,dA$, and $\tfrac1{15}\int K^2\,dA$, where $K$ is <Gaussian curvature>. If its scalar spectrum agrees with that of a <hyperbolic surface> of area $A$, these integrals are $A,-A/3,A/15$. Expanding the displayed square then forces $K=-1$ everywhere. Thus constant negative curvature is spectrally determined in this setting, although the global hyperbolic metric need not be determined up to <isometry>.
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