Background:
Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of thickness $\epsilon>0.$ Let $\lambda_i(T_\epsilon,g)$ denote the Neumann eigenvalues with respect to the Laplacian $\Delta_g$ and $\lambda_i(\gamma,g_{\text{eucl}})$ denote Neumann eigenvalues on the interval $[0,\pi]$ (here $g_{\text{eucl}}$ denotes the euclidean metric on the interval $[0,\pi]$).
Question:
- Is there an explicit form in which I can write the induced metric on $T_{\epsilon}?$ I read about Fermi normal coordinates but I am not sure how to write the metric in fermi normal coordinates around a geodesic on the sphere.
- Is it true that each eigenvalue $\lambda_i(T_\epsilon,g)\to \lambda_i(\gamma,g_{\text{eucl}})$ as $\epsilon\to 0?$
