  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 show it is equivalent to the strong Arnold hypothesis,
which 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 seven, and exhibit examples of metrics violating it.

Differential Geometry;Analysis of PDEs;Spectral Theory