On the infinitesimal orbit type of maximal dimensional orbits

We prove that in the case of an isometric action α:G×M→M of a Lie group G on a semi-Riemannian manifold M the union of the maximal dimensional orbits is an open and dense set in M. Moreover, if M is a Lorentz manifold and α is an isometric action on it, then in the set of the maximal dimensional orbits local stability and normalizability are equivalent, and there is no open invariant set U⊂M such that all the orbits G(x)⊂U are non-normalizable and have the same infinitesimal type. These results are useful in the extension of the principal orbit type theorem.