We establish the Bessel-Geodesic Theorem, proving that the zeros of the geodesic component of the covariant derivative of an exponentially growing Bessel vector field on a curved 2-dimensional submanifold are displaced from their flat-space positions by an amount controlled by the principal curvatures of the submanifold and the sectional curvature of the ambient manifold. Under an exponentially growing force profile, this displacement is exponentially amplified at second order in curvature with explicit geometric coefficient. The flat-space limit is cleanly recovered when all curvatures vanish.
Edward Lendward Smith (Sun,) studied this question.