1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

400 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY

REMARK G.31.

(1) Note that the distance between two points x and y in the sphere

sn-l of radius 1 is the same as the angle LxOy, where 0 is the

origin of ]Rn.

(2) When X = sn-^1 , the metric (G.12) on

Cone (sn-l) ~ JRn,

where the isomorphism is given by polar coordinates, is the same

as the Euclidean metric (by the law of cosines).

In the remainder of this subsection we give another motivation for the

definition of dcone(X) from the Riemannian metric tensor viewpoint. Let

x1, Xz E sn-l and let a~ d (x1, xz). Consider the Euclidean planar triangle

with side-angle-side equal to rl -a-r2 and vertices with polar coordinates

(r1, 0), (0, 0), and (r2, a). Let f3 be the angle of the triangle at the vertex

(r1, 0) and let A be the straight line segment in the plane from (r1, 0) to

(r2, a). For u E [O, a], consider the line segment B emanating from (0, 0)

with angle u and endpoint on the line A. By the law of cosines, L (A) =

)ri + r§ - 2r1r2 cos a. By the law of sines, we have

(G.13)

Hence

(G.14)

sinf3 sin a

rz

sinf3

L(B)

)ri + r§ - 2r1r2 cos a'

sin ~~~~~-= (1f - f3 - u) ~~~~ sin ((3 + u)

L (B) = rl sinf3

sin f3 cos u + cos f3 sin u '

where sinf3 and cosf3 can be determined from (G.13).

Now let (Mn, g) be a Riemannian manifold and define the metric

9Cone ~ r^2 9 + dr^2

on M x (O,oo). Given two points x1,x2 EM, let

'Y: [O, d (x1, x2)] -+ M

be a unit speed minimal geodesic joining x1 to xz. Given r 1 , rz E (0, oo ),

join the two points (xi, ri), (x2, rz) E M x (0, oo) by paths 1 : [O, d (x, y )] -+

M x (0, oo) of the form

1 (u) = ('Y (u), r (u)),

where r: [O,d(x1,x2)]-+ (O,oo) satisfies r(O) = ri and r(d(x1,x2)) = rz.
Tpe length of 1 with respect to 9Cone is given by