Page 61 - Curvature and Homology
P. 61
EXERCISES
If P' moves along the curve C'(t), we have
that is,
dP' = e;dui = (dud) e;.
Thus, dP' is a vectorial 1 -form. Show that dP' may be considered as that vectorial
1-form giving the identity map of An into itself.
Differentiating the relations (*) with respect to uj we obtain
Again, since ei is a function of the parameter t along C'(t),
that is,
The de; (i = 1, ..., n) are vectorial 1-forms. Hence, in terms of the basis
{ei @ duj},
where the cj are the components of de; relative to this basis. Put
Then,
Show that the matrix (8:) defines a map of the tangent space at P' + dP' onto
the tangent space at P'. Consequently, the functions rfj are the coefficients of
connection relative to the natural basis.
B. Orientation
1. Show the equivalence of the two definitions of an orientation for a differenti-
able manifold. Assume that the form ar of 5 1.6 is differentiable.
Hint: Use a partition of unity.
