Spectral determination of hyperbolic curvature on a surface ($\int (K+1{)}^{2}dA=0$)

On a closed two-dimensional Riemannian manifold, the first three integrated scalar heat invariants are $A$, $\frac{1}{3}\int KdA$, and $\frac{1}{15}\int K^{2}dA$, 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.