- SOME RESULTS IN LOCAL GEOMETRY 289
obtained by taking a~ (Y, a/or) and Z ~ Y -a (8/ar). Then (Z, a/or)= 0
and
8
(Y, \i' f) = Y (f) = a or (f) + Z (f).
We claim that
:r (f) = 1 and z (f) = 0.
The corollary will follow from these formulas, because they imply that
(Y, Y' J) = a ~ \ Y, :r)
for all YE Texpp(v)Mn and all VE B(O,c){O}.
To first formula of the claim is proved by the observation:r (f) = ((expPL (.R)) (ro ( expPIB(O,t:))-
1
) = R(r) = 1.To get the second formula, we note that
Z (f) = Z (r o ( expPIB(O,t:))-
1
)= [ ( ( expPIB(O,t:))-
1
) * (Z)] (r)= \ ( ( expPIB(O,t:))-l) * (Z), R).
But
\z, (expPL (.R)) = \ Z, :r) = 0.
By the Gauss lemma, (expPL maps R,1- to (8/8r)1-; since (expPIB(O,t:))*
is an isomorphism, it follows that ((expPIB(O,t:))-^1 )* maps (8/8r)1- to R1-,
hence thatD1.2. Conjugate points. If r.p : Nn ---+ pn is a smooth map between
differentiable manifolds of the same dimension, we say r.p* : TxNn ---+ Tr.p(x) pnis singular if it is not an isomorphism. In this case we say that x E Nn is
a critical point of r.p and that r.p (x) is a critical value of r.p.DEFINITION B.3. A point q E Mn is a conjugate point of p E Mn if
q is a critical value of
expP : TpMn ---+ Mn,
namely, if q = expP (V) for some V E TpMn such that(expPL: Tv (TpMn)---+ Texpp(v)Mn