Everywhere invariant spaces of metrics and isometries (Q1081161)

Let M be a smooth n-dimensional manifold, \({\mathcal D}(M)\) the group of diffeomorphisms of M, G(M) the set of (pseudo-) Riemannian metrics on M, H the group of bijections of G(M) and \({\mathcal D}^*(M)=\{h\in H:\) \(h=\phi^*\), \(\phi\in {\mathcal D}(M)\}\) (here \(\phi^*\) arises from the natural right action of \({\mathcal D}(M)\) on G(M)). If S is a subgroup of H such that the normalizer of S in H contains \({\mathcal D}^*(M)\), then S.g, the orbit of g under S, is everywhere invariant, i.e., all isometries of all the metrics in S.g are contained in the set of diffeomorphisms leaving S.g invariant. The authors further show that a generalized conformal structure on M (in a natural sense) is an everywhere invariant set of metrics. The results are applied to finding the isometries of plane-fronted gravitational waves with parallel rays.