- NOTES AND COMMENTARY 495
Assuming Presumed Theorem I.50 is true, the definition of M ( oo) in
(I.10) is equivalent to the definition of M ( oo) in (I.65); this follows from (I.8)and (I.61). Moreover, by (I.64) we have that dR ([a], [,BJ) ::::; Cd 00 ([a], [,BJ)
for any [a] , [,BJ E M ( oo). Actually, we expect that dR = d 00 •
REMARK I.55. Analogous to the notion of points at infinity for complete
Riemannian manifolds with nonnegative sectional curvature, for simply-
connected complete Riemannian manifolds of nonpositive curvature there
is a metric, called the Tits metric, on the space of 'points at infinity',
which, as a point-set, is also defined by putting an equivalence relation on
the space of rays and taking the quotient. The metric cone over the space
of points at infinity is called the Tits cone.
7. Notes and commentary
A standard reference for some of the comparison geometry in this appen-
dix is the book by Cheeger and Ebin [30], especially Chapter 8 on 'complete
manifolds of nonnegative curvature'. We also refer the reader to the seminal
papers of Cheeger, Gromoll, Meyer, Perelman, and Sharafutdinov, i.e., [32],
[31], [77], [148], and [171].
§1. For the Definition I.21 of soul, seep. 422 of [32] or the beginning of
Chapter 8 of [30]. For a proof of Theorem I.22(1) and (3), see also Theorem
8.11 and Corollary 8.12 of [30].
§2. For Theorem I.26 see Theorem 5.3 in Shiohama [175] or Lemma 3.4
in Guijarro and Kapovitch [87].
§4. Critical point theory for the distance function was first developed
by Grove and Shiohama [83]. Excellent references are Meyer [127], Cheeger
[29], Grove [81], and Karcher [105].
For Lemma I.34, see pp. 360-361 of [81] (see also Lemma 3.6 of [127]
and Proposition 47 on pp. 335-337 of Petersen [155]).^22 For Lemma I.37
see Lemma 1.4(ii) of [107].
(^22) Note that there was a typographical error in the proof of Proposition 1.2 on pp.
320-321 in the first edition of [155] which is corrected in the second edition.
'; \