Let M n denote an «-dimensional complete open Riemannian manifold. In [AG] Abresch and Gromoll introduced a new concept of "diameter growth." Roughly speaking, one would like to measure the essential diameter of ends at distance r from a fixed point p e M n . They showed that M n is homotopy equivalent to the interior of a compact manifold with boundary if M n has nonnegative Ricci curvature and diameter growth of order o(r l / n ), provided the sectional curvature is bounded from below. It is

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... ell known that any complete open manifold with nonnegative sectional curvature has finite topological type. This is a weak version of the Soul Theorem of Cheeger-Gromoll [CG]. Examples of Sha and Yang show that this kind of finiteness result does not hold for complete open manifolds with nonnegative Ricci curvature in general (see [SY1, SY2]), and additional assumptions are therefore required. We will use a concept of the essential diameter of ends slightly stronger than that of [AG]: For any r > 0, let B(p,r) denote the geodesic ball of radius r around p. Let C(p,r) denote the union of all unbounded connected components of M n \B(p,r). For r 2 > r { > 0, set C{p\r u r 2 ) -C(p,ri) n B(p,r 2 ). Let 1 > a > fi > 0 be fixed numbers. For any connected component £ of C{p\ ar, ^r), and any two points x,y e ZndB(p, r), consider the distance d r (x, y) -inf Length() between x and y in C(p, fir), where the infimum is taken over all smooth curves (/ > c C(p, fir) from x to y. Set diam(Zn<9 B(p, r), C(p, fir)) = sup d r (x,y), where x, y etr\dB(p,r). Then the diameter of ends at distance r from p is defined by diam(/?> r) = sup diam (E n dB(p, r), C(p, fir)), where the supremum is taken over all connected components X of C{p\ ar, ^r). The diameter defined here is not smaller than that defined by Abresch and Gromoll. Our definition will be essential in Lemma 3 and its applications. The purpose of this note is to announce the following results. THEOREM A. Let M be a complete open Riemannian manifold with sectional curvature KM > -K 2 for some constant K > 0. Assume that for some base point p e M, limsupdiam(p,r) < -^-.