Page 145 - Curvature and Homology

EXERCISES 127

space at P(t) = exp(tX)P. The induced dual map exp(tX)* sends A(T&)
into A (T;). For an element E /\"(Tp*,t,)

is an element of AP(T,*). Show that

and that consequently

for any elements a, B E A *(T).
Hint: Show that

C. Frobenius' theorem and infinitesimal transformations
1. Show that the conditions in Frobenius theorem (I. D.4) may be expressed
in the following form: If the basis of the tangent space Tp at P E M is chosen
so that the subspace F(P) of Tp of dimension r is spanned by the vectors
XA(A = q + 1, ... , n) then, if we take 8 = e1 A ... A 89 the conditions of
complete integrability are given by

This is equivalent to the condition that [XA,XB] is a linear combination of
Xu+,, .-. , X, only. In other words, F is completely integrable, if and only if,
for any two infinitesimal transformations X,Y such that Xp, Yp' E F(P) for all
P E U the bracket [X, YIP E F(P).
2. Associated with the vector fields X and Y are the local one parameter groups
vX(P,t) and vy(P,t). Then, [X,YIP is the tangent at t = 0 to the curve
t
t
t
t
?)
WX. dP, t ) = vY (vx (?Y (vx (p, T)9 z), 3),
-
-
This formula shows, geometrically, the necessity of the integrability conditions
for F. For, if Xp and Yp are contained in F(P) for all P E U and F is integrable,
the integral curves of X and Y must be contained in the integral manifold.
Hence, the formula shows that the above curves must also be contained in the
integral manifold from which it follows that [X,YJp E F(P).