April 11, 1996
1. This problem is about the tangent bundle to the 2-sphere . As usual, we write , where U is the complement of the South pole and V is the complement of the North pole. Show how the tangent bundle admits trivializations over U and V (i.e. isomorphisms to the product bundles and ) and exhibit the transition function relating those trivializations over .
2. The real projective plane can be considered as , where as usual , and the relation is . The map defined by
defines a map which is invariant under , and therefore defines a map .
a. Show that v is an immersion, i.e. that v has rank 2 at every point of .
b. Show that v is one-one.
c. Suppose v is composed with projection onto by suppressing one of the first three coordinates. Is the resulting map still an immersion? Is it one-one?
3. Let be a smooth vectorfield defined in an open set , and suppose S is a smooth, oriented piece of surface lying in U, with boundary C (C inherits an orientation from S as usual). The ``Traditional Stokes' Theorem'' states that the integral over S of the curl of X is equal to the integral of X along C. [Recall that the integral of a vectorfield X over a piece of surface parametrized by , is given by , and that the integral of X along a curve parametrized by is .] Show that this theorem is equivalent to a special case of our ``Stokes' Theorem.''