Global eigenfamilies on closed manifolds

We study globally defined $(\lambda ,\mu )$-eigenfamilies on closed Riemannian manifolds. Among others, we provide (non-)existence results for such eigenfamilies, examine their topological properties and classify $(\lambda ,\mu )$-eigenfamilies on flat tori. It is further shown that for $f=f_1+i{f}_2$ being an eigenfunction decomposed into its real and its imaginary part, the powers $\{f_1^a{f}_2^b\mid a,b\in \mathbb{N}\}$ satisfy highly rigid orthogonality relations in $L^2(M)$. In establishing these orthogonality relations one is led to combinatorial identities involving determinants of products of binomials, which we view as being of independent interest.