Looping directions and integrals of eigenfunctions over submanifolds

Abstract: Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold without boundary and $e_\lambda$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator with respect to the metric $g$, i.e [ -\Delta_g e_\lambda = \lambda^2 e_\lambda \qquad \text{ and } \qquad | e_\lambda |{L^2(M)} = 1. ] Let $\Sigma$ be a $d$-dimensional submanifold and $d\mu$ a smooth, compactly supported measure on $\Sigma$. It is well-known (e.g. proved by Zelditch in far greater generality) that [ \int\Sigma e_\lambda \, d\mu = O(\lambda^\frac{n-d-1}{2}). ] We show this bound improves to $o(\lambda^\frac{n-d-1}{2})$ provided the set of looping directions, [ \mathcal{L}_{\Sigma} = { (x,\xi) \in SN^*\Sigma : \Phi_t(x,\xi) \in SN^*\Sigma \text{ for some } t > 0 } ] has measure zero as a subset of $SN^*\Sigma$, where here $\Phi_t$ is the geodesic flow on the cosphere bundle $S^*M$ and $SN^*\Sigma$ is the unit conormal bundle over $\Sigma$.