The study of finiteness for Riemannian manifolds, which has been done originally by J. Cheeger [5] and A. Weinstein [13], is to investigate what bounds on the sizes of geometrical quantities imply finiteness of topological types, —e.g. homotopy types, homeomorphism or diffeomorphism classes-— of manifolds admitting metrics which satisfy the bounds. For a Riemannian manifoldMwe denote byRMandKMrespectively the curvature tensor and the sectional curvature, by Vol (M) the volume, and by diam(M) the diameter.