It is well known that compact Riemannian manifolds $(M, g)$ with periodic geodesic flows have ( Besse Book) exceptional spectral properties: the spectrum of $ \sqrt{ - \Delta}$, the square root of the Laplacian, concentrates along the arithmetic progression [$(\frac{ 2 \pi}{T}$) $(k + \beta)$: $k=1, 2, ...$] with $T$ the (minimal) period of the geodesic flow $G^{t}$ and with ; $\beta$ the common Morse index of the $T$-periodic geodesics.
My question is: $\beta$ is a integer number if $T = \pi$, for example ?
