In this paper we consider classes of Riemannian manifolds (M, g) with bounds on the five fundamental geometric invariants: Ricci curvature Ric,L" /2 -norm of curvature / |Rm|" /2 ί/g, injectivity radius inj(g), diameter diam(g) and volume Vol((7).Let G(H, / 0 , K, n) denote the collection of all closed, connected ndimensional Riemannian manifolds (M, g) satisfying |Ric| < H, inj(g) >/ 0 >0,and f M \Rm(g)\ n/2 dg V is precompact among " C lc *-Riemannian manifolds" [17], [12], [27].Our first result says that we have a similar compactness result in G(H < i o , K, ή), which roughly states that: Given a sequence {M k } of compact Riemannian ^-manifolds with | Ric | < H, diam < D, J|Rm|" /2 dg < K, and inj > z' o < 0, then {M k } has a subsequence, away from finite number points, which converges to an ^-manifold M with C lα metric g for an a e (0, 1). The precise statement is the following (n > 4).Theorem 0.1. Let {{M k , g k )} be a sequence of Riemannian manifolds in G{H, i 0 , K, ή), with diam(A/ A: ) < D. Then there exist a subsequence of {{M k , g k )} (by renumbering, we still use {M k , g k }), and a sequence {rj, r ι -• 0 when I -> oc, such that the following hold:(a) There exists a C°°-manifold M, such that M k is diffeomorphic to M for each large k .(b) There exist a C 00 -metric g on M, and finite number points {m f ,-• , m h ] with h < C{H, i 0 , K, n), such that g is a C 1 '°-metric on M -{m { , , m k }, 0 < oc < 1.
