On common zeros of eigenfunctions of the Laplace operator

We consider the eigenfunctions of the Laplace operator $\Delta$ on a compact Riemannian manifold of dimension $n$. For $M$ homogeneous with irreducible isotropy representation and for a fixed eigenvalue of $\Delta$ we find the average number of common zeros of $n$ eigenfunctions. For this we compute the volume of the image of $M$ under an equivariant immersion into a sphere.