Pinching and Collapse

Abstract

The group of diffeomorphisms Diff(V) of a smooth manifold V naturally acts on the space of Riemannian metrics g on V; various classes of metrics one studies in geometry are usually invariant under Diff(V). In fact, we tend not to distinguish isometric manifolds, and a diffeomorphism f : V → V establishes an isometry between (V, g) and (V, f*(g)) for each metric g. Furthermore, the geometric dictum “from local to global” suggests the study of locally defined classes of metrics on V which are moreover Diff-invariant.