The classical eigenproblem focuses on eigenvalues and eigenvectors of linear operators acting on a vector space. The matrix representation of the problem has been extended towards multidimensional arrays, with various applications. Another extension addresses invariant subspaces of multilinear operators in Banach spaces. The generalization of the eigenproblem for tensors is still a challenging issue. We investigate eigenproblems of supersymmetric tensors on Riemannian manifolds, emerging from the initial proper definitions, with the proposed extensions.