- BUSEMANN FUNCTIONS 303
of constant sectional curvature H with the same side lengths (a, b, c) such
that
o: 2 a~ LrpiJ.
f3 2 jJ ~ Lpqr.Hinge version (SAS): Let L be a geodesic hinge with vertices (p,q,r),
sides qr and rp, and interior angle Lqrp E [O, 7r] in Mn. Suppose that qr is
minimal and that length ( rp) :S 7r /Vii if H > 0. Let L^1 be a geodesic hinge
with vertices (p', q', r') in the complete simply-connected space of constant
sectional curvature H with the same side lengths and same angle. Then one
may compare the distances between the endpoints of the hinges as follows:
dist (p, q) :S dist (p', q').
- Busemann functions
In this section, we develop some tools to study a complete noncompact
Riemannian manifold (Mn, g) at very large length scales, in order to under-
stand its geometry 'at infinity'.
3.1. Definition and basic properties.
DEFINITION B.41. A unit speed geodesic 'Y : [O, oo) -+ Mn is a ray ifeach segment is minimal, namely, if 'Yl[a,b] is minimal for all 0 :S a< b < oo.
We say 'Y is a ray emanating from 0 E M if 'Y is a ray with 'Y (0) = 0.
LEMMA B.42. For any point 0 E Mn, there exists a ray 'Y emanating
from 0.PROOF. Choose any sequence of points Pi E Mn such that d ( 0, Pi) /
oo. For each i, choose a minimal geodesic segment 'Yi joining 0 and Pi· LetVi= d"fif dt (0) E ToMn.
Since the unit sphere in ToMn is compact and !Vil
subsequence such that the limit
(B.4) V ~ .lim Vi1, there exists aexists. Let 'Y : [O, oo) -+ Mn be the unique unit-speed geodesic with "( (0) =
0 and d"f/dt (0) = V. Recalling that the solution of an ODE is a continuous
function of its initial data, we note that (B.4) implies the images 'Yi -+ "(
uniformly on compact subsets of Mn. Because each segment of every 'Yi is
minimal, it follows that each segment of 'Y is minimal. DDEFINITION B.43. If 'Y is a ray emanating from 0 E Mn, the pre-
Busemann function b 1 ,s : Mn -+ JR associated to the ray "( and s E [O, oo)
is defined by
b 1 ,s (x) ~ d (0, 'Y (s)) - d ("! (s), x) = s - d ("! (s), x).