On the space of Riemannian metrics

Abstract

1. Results. In the present announcement we are concerned with the space of Riemannian metrics on a compact smooth manifold. Let M be such a manifold, 5T* the bundle of symmetric covariant twotensors on Mf and C°°(S T*) the smooth sections of this bundle, endowed with the C topology. If 9filCC°°(5r*) is the set of smooth Riemannian metrics on M (those sections which at each point p of M induce a positive definite bilinear form on Tp, the tangent space to M), it is well known that 9II is an open convex cone in C^iS^T*). If 3D is the group of diffeomorphisms of M (with the C°° topology), 3D acts on C*(S*T*) on the right by pull-back and 3TC is an invariant set under the action. We write A : 3 ) X C ° ° ( 5 2 r * ) ^ ^ ( 5 2 r * ) and denote A fa, 7) by y*(y)' A is a right action because (£rc)*Y ==??*£*(7). Now restrict to A: £>X9TC->9TC. For any X£3TC define Jx, the isotropy group of X, by J = {T?E3D|T7*(X) =X}. For a fixed yC~?iil, let Oy be the orbit of 3D through 7.