ON THE COMPACTNESS OF A CLASS OF RIEMANNIAN-MANIFOLDS

A class of Riemannian manifolds is studied in this paper. The main conditions are 1) the injectivity is bounded away from 0; 2) a norm of the Riemannian curvature is bounded; 3) volume is bounded above; 4) the Ricci curvature is bounded above by a constant divided by square of the distance from a point. Note the last condition is scaling invariant. It is shown that there exists a sequence of such manifolds whose metric converges to a continuous metric on a manifold.