1547671870-The_Ricci_Flow__Chow

l. SOME RESULTS IN LOCAL GEOMETRY 293

where a, b E IR, and Jo is a Jacobi field such that

(Jo, T) = O.

PROOF. Since 'Y is a geodesic, we have \lrT = 0 and hence

-d d2 I

t^2 (J, T) t=O = (\lr\lrJ, T) = (R (T, J) T, T) = 0.

D

1.4. Focal points and the normal bundle. Focal points are a nat-
ural generalization of the notion of conjugate points.

DEFINITION B.15. If 1,: r,k q.._, Mn is an immersion of a smooth manifold
r,k, the normal bundle of r,k is the sub bundle NL,k of the pullback bundle
1,*T Mn whose fiber over each p E r,k is

NPL,k = {VE TpMn: (V, W) = 0 for all WE 1,* ( Ti-l(p)r,k)}.

Consider

expPIN p L;K : NpL,K----+ Mn.

Taking the union over all p E r,K, one obtains a map of the total space

explNL; : NL,k ----+ Mn

defined for all pairs (p, V) with p E r,k and V E NpL,k by

explNL;k (p, V) ~ expPIN p L;k (V).

Notice that explNL;k : NL,k ----+ Mn is a smooth map between manifolds of
the same dimension.

EXAMPLE B.16. If r,n = Mn, then NMn = Mn x {O} is naturally

identified with Mn, so that we may regard exp[NMn: NMn----+ Mn as the

identity map.

EXAMPLE B.17. If r,o = {p} is a point, then T {p} = {0}, so that

N {p} = TpMn and explN{p} = expp.

DEFINITION B.18. Let 1,: r,k q.._, Mn be an immersion.. One says q E Mn

is a focal point of 1, (r,k) if q is a critical value of explNL;k· In particular,

one says q is a focal point of r,k at p if it is the image of some critical point

in NPL,k.

Notice that q E Mn is a focal point of r,k ~ Mn only if q ~ r,k.

EXAMPLE B .19. Let Mn= IRn and let r,n-l = s;_i- l (p) be the (n - 1)-

sphere of radius r > 0 centered at p E Mn. Then the unique focal point of

r,n-l is p.