On metric and algebraic properties of spherical harmonics

The eigenfunctions of the Laplace operator on spheres are called the spherical harmonics. They can be characterized as the traces of homogeneous harmonic polynomials onto the spheres. We'll consider bounds for the volumes of their nodal sets (i.e., the sets of all zeros), common zeros, and their divisors.
On a compact connected Riemannian manifold whose first de Rham cohomologies are trivial every pair of eigenfunctions has a common zero if they correspond to the same eigenvalue. It is possible to find bounds for volumes of the nodal sets and some other metric quantities related to the spherical harmonics using methods of Integral Geometry.
Every equivariant mapping of an isotropy irreducible homogeneous space is a local metric homothety. This makes it possible to calculate mean volumes of level sets of random linear combinations of the eigenfunctions using the natural mappings to the spheres.
The spherical harmonics are connected with the Legendre polynomials. There are series of quadratic divisors of the associated Legendre polynomials. On the other hand, only a finite number of polynomials of degree 3 and 4 can be divisors.