Closed manifolds with small excess

In this article, we study closed Riemannian manifolds with small excess. We show that a closed connected Riemannian manifold with Ricci curvature and injectivity radius bounded from below is homeomorphic to a sphere if it has sufficiently small excess. We also show that a closed connected Riemannian manifold with weakly bounded geometry is a homotopy sphere if its excess is small enough.