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Abstract.In this note we study the following problem: When must a complete Einstein metric g on an «-manifold with Ric = (n -X)Xg be a constant curvature metric of sectional curvature I ?

We show that if a complete open manifold with bounded curvature and sufficiently small ends, then each end is an infranilend. Conversely, an open manifold with finitely many infranilends admits a complete metric with bounded curvature and arbitrarily small ends.

Abstract.In this paper we establish some vanishing and finiteness theorems for the topological type of complete open riemannian manifolds under certain positivity conditions for curvature. Key tools are comparison techniques and Morse Theory of Busemann and distance functions.