  This paper is concerned with the structure of the set of Riemannian metrics
on a connected manifold such that the corresponding Laplace-Beltrami operator
has an eigenvalue of a given multiplicity. We introduce a simple geometric
condition on metrics and their corresponding Laplace eigenfunctions, and show
it is equivalent to the strong Arnold hypothesis. This hypothesis essentially
posits that multiple Laplace eigenvalues split up under perturbation like those
of symmetric matrices. We prove our condition is satisfied except on a set of
infinite codimension, and use this to obtain non-crossing rules for Laplace
eigenvalues. Furthermore, we show that our condition is satisfied on all
metrics admitting eigenvalues of multiplicity at most six, and exhibit examples
of metrics violating it.
