The accumulated distribution of quadratic forms on the sphere (Q1923202)

Let \(A\) be a real symmetric matrix of size \(N\), \(q\) the associated quadratic form \(q(v)= v^TAv\) restricted to the unit sphere \(S^{N-1}\) in \(R^N\), and \(Q(t)\) the accumulated distribution of \(q\): \(Q(t)= {1\over\text{vol} (S^{N-1})} \int_{D_t} dV\), where \(dV\) is the standard volume form on the unit sphere \(S^{N-1}\) and \(D_t= \{v\in S^{N-1} \mid q(v)< t\}\). The authors study the smoothness properties of \(Q\) and give a detailed description of its behavior at the eigenvalues of \(A\), from which it is shown that \(Q\) fails to be analytic at such points. The result follows from an argument full of calculations, which is split into steps.