 
Summary: RESEARCH BLOG 7/24/03
We want to see the relation between noncollapsing and injectivity
radius, to understand Corollary 4.2 of [2]. For some of the notation,
see the previous research blog 7/23/03
Lemma 0.1. Given constants > 0, > 0, K > 0, and n N, there
is a constant c such that if (M, g) (a compact Riemannian nmanifold)
is noncollapsed on the scale and Rm(g) K, then
inj(M)2
Rm(g) c.
Proof. Suppose that we have a compact Riemannian manifold (M, g)
which is noncollapsed on the scale . By rescaling g by Rm(g)
(and relabelling this metric g), we may assume that the manifold sat
isfies Rm(g) = 1, and (M, g) is noncollapsed on the scale K
1
2 (see
the comment after Def. 4.2 of [2]). Let be the length of a minimal
geodesic in (M, g), and assume < 2K
1
2 . Since we have assumed
Rm(g) = 1, by theorem 3.4, [1], inj(M, g) min{, 1
