1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

350 K. IMPLICIT FUNCTION THEOREM

second fundamental form of sn equals the first fundamental form of sn. Our sign

conventions are such that for vector fields X and Y on sn,

(K.56) Dxv=ll(X)=X

and

(K.57) Dx Y = \7 x Y - II (X, Y) v = \7 x Y - (X, Y) v.

Now extend v to ffi.n+l - {O} as v (x) = l~I so that Dvv = 0. Given a vector

field X on sn, extend X to ffi.n+I to be homogeneous of degree 1. In particular,

X (x) = JxJ X( l~I) for x E ffi.n+l - {O}. We then have DvX = X and

[X, v] = Dxv - DvX = X - X = 0.

We have the following.

LEMMA K.25 ( o on sn in terms of o on ffi.n+I). If T/ is a p-form defined on a

neighborhood of sn in ffi.n+I' then

( Osn T/lsn) (X1, ... , Xp-1) - ( OJRn+1 T/) (X1, ... , Xp-1)

= (n - 2p + 2) T/ (v, X1,... , Xp-1) + v (ry (v, X1, ... , Xp-1)),

Where the vector fields Xj on sn are extended to ffi.n+l to be homogeneous of degree

1, so that DvXj = Xj.

PROOF. Recall that on a Riemannian manifold (Mn,g) we have that o
QP (M) --t QP-^1 (M) is given by

n

(ory) (X1,... ,Xp-1) = -L (VeiT/) (ei,X1, ... ,Xp-1)

i=l

for all vector fields X1, ... ,Xp_ 1 , where {ei}~=l is a local orthonormal frame field.

On sn we have by the product rule and by (K.57) that for each i,

(DeJJ) (ei,X1, .. · ,Xp-1) - (\lei T/Jsn) (ei,X1, ... ,Xp-1)

= T/ (\7 ei ei - Deiei, X1, ... , Xp-1)

p-1

+LT/ (ei, X1, ... , VeiXJ - DeiXj,... , Xp-1)

j=l

= T/ (v, X1, ... , Xp-1)

p-1

+ (ei, Xj) LT/ (ei, X1, ... 'xj-1, v , Xj+1, ... 'Xp-1).

j=l