On common zeros of eigenfunctions of the Laplace operator (Q525197)

The Laplace operator and zeroes on its eigenfunctions for compact Riemannian homogeneous spaces \(M^n\) with irreducible isotropy are considered. The main object is the set of common zeroes for eigenfunctions with fixed eigenvalue. The average number of such common zeroes is found (the average is taken over the Grassmanian \(\mathrm{Gr}_n\), with respect to the action of \(\mathrm{SO}(N,\mathbb R)\)). It is done by computing the volume of the image of \(M\) under an equivariant immersion into a sphere \(S^{N-1}\) and using the kinematic formula from integral geometry.