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\begin{document}$$(M,\,g)$$\end{document} be a compact n-dimensional Riemannian manifold without boundary and eλ\documentclass[12pt]{minimal}
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\begin{document}$$e_\lambda $$\end{document} be an L2\documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document}-normalized eigenfunction of the Laplace–Beltrami operator with respect to the metric g, i.e., -Δgeλ=λ2eλandeλL2(M)=1.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} -\Delta _g e_\lambda = \lambda ^2 e_\lambda \quad \text {and} \quad \left\| e_\lambda \right\| _{L^2(M)} = 1. \end{aligned}$$\end{document}Let Σ\documentclass[12pt]{minimal}
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\begin{document}$$\varSigma $$\end{document} be a d-dimensional submanifold and dμ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{d}\mu $$\end{document} a smooth, compactly supported measure on Σ.\documentclass[12pt]{minimal}
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\begin{document}$$\varSigma .$$\end{document} It is well known (e.g., proved by Zelditch, Commun Partial Differ Equ 17(1–2):221–260, 1992 in far greater generality) that ∫Σeλdμ=Oλn-d-12.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \int _\varSigma e_\lambda \, \mathrm{d}\mu = O\left( \lambda ^\frac{n-d-1}{2}\right) . \end{aligned}$$\end{document}We show this bound improves to oλn-d-12\documentclass[12pt]{minimal}
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\begin{document}$$o\left( \lambda ^\frac{n-d-1}{2}\right) $$\end{document} provided the set of looping directions, LΣ={(x,ξ)∈SN∗Σ:Φt(x,ξ)∈SN∗Σfor somet>0}\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\mathcal {L}}}_{\varSigma } = \{ (x,\,\xi ) \in \mathrm{SN}^*\varSigma : \varPhi _t(x,\,\xi ) \in \mathrm{SN}^*\varSigma \text { for some } t > 0 \} \end{aligned}$$\end{document}has measure zero as a subset of SN∗Σ,\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{SN}^*\varSigma ,$$\end{document} where here Φt\documentclass[12pt]{minimal}
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\begin{document}$$\varPhi _t$$\end{document} is the geodesic flow on the cosphere bundle S∗M\documentclass[12pt]{minimal}
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\begin{document}$$S^*M$$\end{document} and SN∗Σ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{SN}^*\varSigma $$\end{document} is the unit conormal bundle over Σ.\documentclass[12pt]{minimal}
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\begin{document}$$\varSigma .$$\end{document}
