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We consider the wave equation on a closed Riemannian manifold. We observe therestriction of the solutions to a measurable subset $\omega$ along a timeinterval $[0, T]$ with $T>0$. It is well known that, if $\omega$ is open and ifthe pair $(\omega,T)$ satisfies the Geometric Control Condition then anobservability inequality is satisfied, comparing the total energy of solutionsto their energy localized in $\omega \times (0, T)$. The observability constant$C\_T({\omega})$ is then defined as the infimum over the set of all nontrivialsolutions of the wave equation of the ratio of localized energy of solutionsover their total energy. In this paper, we provide estimates of theobservability constant based on a low/high frequency splitting procedureallowing us to derive general geometric conditions guaranteeing that the waveequation is observable on a measurable subset $\omega$. We also establish that,as $T\rightarrow+\infty$, the ratio $C\_T({\omega})/T$ converges to the minimumof two quantities: the first one is of a spectral nature and involves theLaplacian eigenfunctions, the second one is of a geometric nature and involvesthe average time spent in $\omega$ by Riemannian geodesics.

Observability properties of the homogeneous wave equation on a closed manifold