The initial value problem for cohomogeneity one Einstein metrics
If a compact Lie group \(G\) acts with cohomogeneity one on a manifold \(M\) (i.e. an action whose principal orbits are hypersurfaces in \(M)\), then a smooth \(G\)-invariant Einstein metric \(g(\text{Ric} (g)=\lambda g)\) with any Einstein constant \(\lambda\) is obtained in a tubular neighborhood around a singular orbit \(Q\) for any prescribed \(G\)-invariant metric \(g_Q\) and second fundamental form \(L_Q\) on \(Q\), provided the representations of the principal isotropy group on the tangent and normal space of \(Q\) have no common subrepresentations. Examples show that \(g\) is not uniquely determined by \(g_Q\) and \(L_Q\). The proof involves ODE techniques and representation theory of the principal and singular isotropy group.
scientific article; zbMATH DE number 1542070
scientific article