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We consider the inverse problem to determine a smooth compact Riemannianmanifold $(M,g)$ from a restriction of the source-to-solution operator,$\Lambda_{\mathcal{S,R}}$, for the wave equation on the manifold. Here,$\mathcal{S}$ and $\mathcal{R}$ are open sets on $M$, and$\Lambda_{\mathcal{S,R}}$ represents the measurements of waves produced bysmooth sources supported on $\mathcal{S}$ and observed on $\mathcal{R}$. Weemphasise that $\overline{\mathcal{S}}$ and $\overline{\mathcal{R}}$ could bedisjoint. We demonstrate that $\Lambda_{\mathcal{S,R}}$ determines the manifold$(M,g)$ uniquely under the following spectral bound condition for the set$\mathcal{S}$: There exists a constant $C>0$ such that any normalizedeigenfunction $\phi_k$ of the Laplace-Beltrami operator on $(M,g)$ satisfies  \begin{equation*}  1\leq C\|\phi_k\|_{L^2(\mathcal{S})}.  \end{equation*}  We note that, for the Anosov surface, this spectral bound condition isfulfilled for any non-empty open subset $\mathcal{S}$. Our approach is based onthe paper [18] and the spectral bound condition above is an analogue of theHassell-Tao condition there.

Disjoint data inverse problem on manifolds with quantum chaos bounds