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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., $$begin{aligned} -Delta _g e_lambda = lambda ^2 e_lambda quad text {and} quad left| e_lambda right| _{L^2(M)} = 1. end{aligned}$$Let (varSigma  be a d-dimensional submanifold and (mathrm{d}mu  a smooth, compactly supported measure on (varSigma .) It is well known (e.g., proved by Zelditch, Commun Partial Differ Equ 17(1–2):221–260, 1992  far greater generality) that $$begin{aligned} int _varSigma e_lambda , mathrm{d}mu = Oleft( lambda ^frac{n-d-1}{2}right) . end{aligned}$$We show this bound improves to (oleft( lambda ^frac{n-d-1}{2}right)  provided the set of looping directions, $$begin{aligned} {{mathcal {L}}}_{varSigma  =  (x,,xi ) in mathrm{SN}^*varSigma : varPhi _t(x,,xi ) in mathrm{SN}^*varSigma text { for some  t u003e 0 } end{aligned}$$has measure zero as a subset of (mathrm{SN}^*varSigma ,) where here (varPhi _t) is the geodesic flow on the cosphere bundle (S^*M) and (mathrm{SN}^*varSigma  is the unit conormal bundle over (varSigma .)

Looping Directions and Integrals of Eigenfunctions over Submanifolds