1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

402 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY

Ricci fl.ow, spaces of less regularity arise when we take limits of smooth so-

lutions to the Ricci flow without a uniform lower bound on the injectivity

radius, i.e., when collapse occurs. A natural collection of spaces with less

regularity, well-suited for the study of Ricci flow and more generally Rie-

mannian geometry, are Aleksandrov spaces with curvature bounded from

below.

In this section, only for the sake of the convenience of the reader and

borrowing especially from [18], we recall some basic definitions and results

on Aleksandrov spaces.

2.1. Motivation.

Returning to the (pre )compactness of a class of Riemannian manifolds,

it turns out that for a complete length space which is a Gromov-Hausdorff

limit of a sequence of Riemannian manifolds with a uniform lower or upper

sectional curvature bound, although its curvature may not be well defined,

the notion of a lower or upper bound of curvature may still make sense.^14

A reason for this is the Toponogov comparison theorem for complete Rie-

mannian manifolds with a lower sectional curvature bound, which we now

recall.

THEOREM G.33 (Toponogov comparison theorem). Let (Mn,g) be a

complete Riemannian manifold with sectional curvatures bounded below by

kER

(1) Triangle version. Let .6. be a triangle in M with vertices (p, q, r)

and sides qr, rp, pq which are geodesics^15 and whose lengths satisfy the

triangle inequality:

L (qr) :SL (rp)+L (pq), L (rp) :SL (qr)+L (pq), L (pq) :SL (qr)+L (rp).

Let Lrpq, Lpqr, Lqrp E [O, 7r] denote the interior angles and assume that

L (pq) :S 7r/Vk if k > 0.

If the geodesics qr and rp are minimal, then there exists a geodesic trian-

gle LS. = (p, q, r) in the complete simply-connected surface of constant Gauss

curvature k with the same side lengths

such that

L (ii.r) = L (qr)' L (rfi) = L (rp)' L (fiii.) = L (pq)

Lrpq ?::: Lrpq,

Lpqr ?::: Lpqr.

We call LS. the k-comparison triangle of .6..

(^14) We shall primarily be interested in spaces with a lower bound for the curvature.
(^15) Such a triangle is not necessarily a 'geodesic triangle', which is defined for the more
general setting of Aleksandrov spaces in subsection 2.2 below.