Page 63 - Curvature and Homology

EXERCISES 45
2. The algebra E can be realized as T(V)/Ze, where T(V) is the tensor algebra
over V and I, is the ideal generated by the elements of the form x @ x, x E V.

D. Frobenius' theorem [23]
The ensuing discussion is purely local. To begin with, we operate in a
neighborhood of the origin 0 in R". Let 8 be a 1-form which is not zero at 0.
The problem considered is to find conditions for the existence of functions f
and g such that
8 = fdg,
that is, an integrating factor for the differential equation

is required. If 8 = fdg, then f(0) # 0. Thus, d8 = df A dg = df A 8/f or

df
dB = o A 8 where w = -*
f
Hence,
8 A do = 0.
Observe that if 8 = fdg, the equation 8 = 0 implies & = 0 and conversely.
Consequently, the solutions or integral surfaces of 8 = 0 are the hypersurfaw
g = const.
As an example, let n = 3 and consider the 1-form

where (x, y, z) are rectangular coordinates of a point in Rs. Then, d8 = y di A dx
+ x d.z A dy. It follows that d8 = dz/z A 8. However, o = di/z is singular
along the z-axis. To avoid this, we may take o = - y dx - x dy. The function
g may be determined. by employing the fact that the integral surfaces g = const.
are cut by the plane x = at, y = bt in the solution z of g(0, 0, z) = const.
On this plane, the equation 8 = 0 becomes

The solution of this ordinary differential equation with the initial condition
40) = c is
2 = ce-aq