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RESEARCH BLOG 7/24/03 We want to see the relation between non-collapsing and injectivity
 

Summary: RESEARCH BLOG 7/24/03
We want to see the relation between non-collapsing 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 n-manifold)
is -non-collapsed 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 non-collapsed 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 non-collapsed 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

  

Source: Agol, Ian - Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago

 

Collections: Mathematics