On the manifolds of eigenvectors of linear and quasilinear finite-dimensional self-adjoint operators. I (Q2754795)

A quasi-linear eigenvalue problem \(A(u)u=\lambda u\), \(u\in S^{n-1}\), where \(A:\;S^{n-1}\to L^{(n)}\) is a smooth mapping (\(L^{(n)}\) is the set of real self-adjoint operators on \(\mathbb R^n\)) prompted the author to study topological properties of the manifold NEWLINE\[NEWLINE P=\{ p=(A,u)\in L^{(n)}\times S^{n-1}:\;\exists \lambda \;\text{for which }Au=\lambda u\} NEWLINE\]NEWLINE and the fibre bundles corresponding to the natural projections of \(P\) to \(L^{(n)}\) and \(S^{n-1}\). They are stratified by the indices and multiplicities of the eigenvalues \(\lambda\). Homotopy properties of the strata are investigated.