For a smooth, closed Riemannian 2-manifold $M \subset \mathbb{R}^3$, is there a relationship between the geodesics of $M$ and smooth functions $k : M \times M \to \mathbb{R}$ which can be expressed in terms of outer products of eigenfunctions of the Laplace-Beltrami operator $\Delta_M$?
Let $(\lambda_j, \phi_j)$ be eigenpairs of $\Delta_M$, so that $$ -\Delta_M \phi_j = \lambda_j \phi_j. $$ Then define $$ k(x_1, x_2) = \sum_j \alpha(\lambda_j) \phi_j(x_1) \overline{\phi_j(x_2)} $$ for some nonnegative function $\alpha : \mathbb{R} \to [0, \infty)$. Let $d(x_1, x_2)$ be the geodesic distance between $x_1$ and $x_2$. Is it true that $k$ is "stationary in geodesic distance" in the sense that for all $x_1, x_2 \in M$ we have $k(x_1, x_2) = f\big(d(x_1, x_2)\big)$ for some $f : \mathbb{R} \to \mathbb{C}$ ?
For closed curves $\Gamma$ (i.e. 1-manifolds) of length $2\pi$, I believe these notions are equivalent. Let $s$ and $t$ denote arclengths on $\Gamma$. Here the Laplace-Beltrami operator is simply the Laplacian, whose eigenfunctions are $e^{ijs}$. So any continuous function $k: \Gamma \times \Gamma \to \mathbb{R}$ that can be written as \begin{align} k\big(x(s), x(t)\big) &= \sum_{j=-\infty}^\infty \alpha(j^2) e^{ijs} \overline{e^{ijt}} \\ &= \sum_{j=-\infty}^\infty \alpha(j^2) e^{ij(s-t)} \\ &= f(s-t) \end{align} is stationary in geodesic distance.
I also believe that these notions are equivalent for the sphere $S^2$. Let $\theta$ be the great circle distance between $x_1$ and $x_2$. Here the eigenfunctions of the Laplace-Beltrami operator are spherical harmonics $Y_{jm}$, and it can be shown using some Bessel function identities [1, Section 3] that \begin{align} k(x_1, x_2) &= \sum_{j=0}^\infty \sum_{m=-j}^j \alpha\big(j(j+1)\big) Y_{jm}(x_1) \overline{Y_{jm}(x_2)} \\ &= \sum_{j=0}^\infty \frac{2j+1}{4\pi} \alpha\big(j(j+1)\big) P_j(\cos \theta) \\ &= f(\theta) \end{align} is stationary in geodesic distance, where $P_j$ is a Legendre polynomial.
So my question is - does this phenomenon generalize to arbitrary smooth manifolds? or is it a coincidence of constant curvature?
My only idea so far is to look at geodesics as a surface heat flow problem, whose solution is governed by the Laplace-Beltrami operator... Any references or suggestions are appreciated!
- Lang, Annika, and Christoph Schwab. "Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations." (2015): 3047-3094.