1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

408 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY

2.4.1. The tangent cone of Aleksandrov spaces with curvature bounded

from below.

Let X be an Aleksandrov space of curvature 2 k, where k E JR., and

let p EX. On the space Sp of shortest paths emanating from p, the (well-

defined) angle Lp at p defines a pseudo-metric. Let ~~ denote the space of

equivalence classes of shortest paths emanating from p, where two shortest

paths are equivalent if the angle between them at p is zero. Then (~~' L.~),

where L.~ is the quotient metric, is a metric space.

The (metric) space of directions ~P is defined to be the metric space

completion of (~~' L.~). We may take the Euclidean metric cone Cone (~p)

over L:p. Recall that we also have the notion of tangent cone defined in

(G.10).

A point p E Xis a regular point if ~P is isometric to sn-l (1). Oth-

erwise, we say that p is a singular point.

We have the following (see 'Theorem 10.9.3, Corollary 10.9.5, and Corol-

lary 10.9.6, all in [18]).

THEOREM G.49 (Existence of the tangent cone). If Xis an n-dimensional

Aleksandrov space of curvature 2 k and p EX, then we have the following:

(1) The tangent cone TpX at p exists^20 and is an n-dimensional Alek-

sandrov space of nonnegative curvature.

(2) TpX is isometric to Cone (~p).^21

(3) When n 2 3, the space of directions ~P is an (n - l)-dimensional

compact^22 Aleksandrov space of curvature 2 1; hence, by Theorem

G.46, ~P has diameter ::; 1f (if n = 1, then the space of directions

is either one or two points; if n = 2, then diam (~p) ::; 1f).

(4) If the space of directions ~Pis isometric to the unit sphere sn-l (1),

then there exists a neighborhood of p homeomorphic to an open set

in JR.n (see p. 13 of[l 75]).

With the above theorem, we may define the exponential map on an

Aleksandrov space with curvature bounded from below (seep. 68 of [175]).

DEFINITION G.50 (Exponential map and cut locus). Let X be an n-

dimensional Aleksandrov space of curvature 2 k and let p E X.

(i) We define Dp c Cone (:Ep) to consist of the set of equivalence classes

[(v, t)] E (~p x lR.;:::o) Irv

such that there exists a geodesic 'Y from p to some point x such

that 'Y is in the direction of v and t = L ( 'Y).

(ii) We define the exponential map

expP: Dp-+ X

(^20) By definition the tangent cone is unique if it exists.
(^21) That is, the tangent cone TpX at p is isometric to the metric cone over the Alek-
sandrov space ~P of curvature ?: 1 (see part (3)).
(^22) See p. 69 of [175].