Einstein Spaces Admitting a One-Parameter Group of Conformal Transformations

Publisher Summary This chapter considers M to be connected complete Einstein space of dimension n > 2 and of class C ∞ and suppose that a vector field on M generates globally a one parameter group of non-homothetic conformal transformations. Then M is isometric to a simply connected space of positive constant curvature. In particular M is homeomorphic to the sphere S n . If M is an irreducible and complete Riemannian manifold of class C ∞ , then A(M) is equal to I(M), except the case M is the one-dimensional euclidean space, where A(M) (I (M)) is the group of a fine (isometric) transformations of M. If a simply connected complete Riemannian manifold of class C ∞ admits a one-parameter group of non-isometric homothetic transformations, then it is a euclidean space.