Geodesic bi-angles and Fourier coefficients of restrictions of eigenfunctions

This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions ${\varphi }_j$ of a compact Riemannian manifold to a submanifold $H\subset M$. We fix a number $c\in (0,1)$ and study the asymptotics of the thin sums, $$N_{ϵ,H}^c(\lambda ):=\sum _{j,{\lambda }_j\le \lambda }\sum _{k:\mathrm{\mid }{\mu }_k-c{\lambda }_j\mathrm{\mid }<ϵ}{\mid {\int }_H{\varphi }_j\overline{{\psi }_k}d{V}_H\mid }^2$$ where $\{{\lambda }_j\}$ are the eigenvalues of ${\sqrt{-\mathrm{\Delta }}}_M,$ and $\{({\mu }_k,{\psi }_k)\}$ are the eigenvalues, resp. eigenfunctions, of ${\sqrt{-\mathrm{\Delta }}}_H$. The inner sums represent the `jumps’ of $N_{ϵ,H}^c(\lambda )$ and reflect the geometry of geodesic c-bi-angles with one leg on $H$ and a second leg on $M$ with the same endpoints and compatible initial tangent vectors $\xi \in S^c_H M, \pi_H \xi \in B^* H$,where$\pi_H \xi$istheorthogonalprojectionof$\xi$to$H$\mathrm{.}Ac-bi-angleoccurswhen$\frac{|\pi_H \xi|}{|\xi|} = c$. Smoothed sums in ${\mu }_k$ are also studied, and give sharp estimates on the jumps. The jumps themselves may jump as $ϵ$ varies, at certain values of $ϵ$ related to periodicities in the c-bi-angle geometry. Subspheres of spheres and certain subtori of tori illustrate these jumps. The results refine those of the previous article (arXiv:2011.11571) where the inner sums run over $k:\mathrm{\mid }\frac{{\mu }_k}{{\lambda }_j}-c\mathrm{\mid }\le ϵ$ and where geodesic bi-angles do not play a role.