1547671870-The_Ricci_Flow__Chow

288 B. SOME RESULTS IN COMPARISON GEOMETRY

Here vtv, Ww E Tw (TpMn) are the vectors identified with V, WE TpMn

respectively, and

(expp)*: Tw (TpMn)-) Texpp(tV)Mn.

Let E be as above. The radial function

r: B(O,E)-) [0,E)

given by

r (V) ~ IVI

defines a function

where

f (x) ~ r ( ( expPIB(O,c:))-l (x)).

The observation f ( expP V) = IVI shows that f is just the distance function

induced by g on expP ( B(O, E)). The function f is smooth everywhere except

at p, so that \J f exists on expP ( B(O, E)) \ {p }.

On the other hand, there is a unit radial vector field R defined on

TpMn\{O} by

Rv - ~ TVT. v

Define a vector field o /or on expP ( B ( 0, E)) \ {p} by

(:) ~ (expPL(.Rv).

r expp(V)

Since

we have

1:r1=1.

The Gauss Lemma implies the following result.

COROLLARY B.2. On expP ( B(O, E)) \ {p}, the vector fields \J f and o /or

are identical:

0

\Jf =or.

PROOF. For every VE B(O,E){O} and every YE Texpp(v)Mn, there is

a unique orthogonal decomposition

0

Y=a 0 r +z