  We prove a trace formula for three-dimensional spherically symmetric
Riemannian manifolds with boundary which satisfy the Herglotz condition: The
wave trace is singular precisely at the length spectrum of periodic broken
rays. In particular, the Neumann spectrum of the Laplace--Beltrami operator
uniquely determines the length spectrum. The trace formula also applies for the
toroidal modes of the free oscillations in the earth. We then prove that the
length spectrum is rigid: Deformations preserving the length spectrum and
spherical symmetry are necessarily trivial in any dimension, provided the
Herglotz condition and a generic geometrical condition are satisfied. Combining
the two results shows that the Neumann spectrum of the Laplace--Beltrami
operator is rigid in this class of manifolds with boundary.
