### Final Exam

This exam will be due at noon on Thursday 19 December.

You are free to use any books or notes, and to ask me questions. But do not discuss these problems with anyone else.

1. Suppose Mn is a closed manifold (that is, compact without boundary). Show that there is no immersion of M into Rn.
2. Suppose {v1 , ... , vk} and {w1 , ... , wk} are two linearly independent sets of vectors in a vectorspace V.  Show that    span{vi} = span{wi}    if and only if    v1 ^ ... ^ vk = c w1 ^ ... ^ wk    for some nonzero constant c.
3. Suppose Mn is an oriented Riemannian manifold with volume form w, and X1 , ... , Xn and Y1 , ... , Yn are vectorfields on M. Show that the product  w(X1 , ... , Xn)  w(Y1 , ... , Yn)  equals the determinant of the n-by-n matrix <Xi , Yj>.
4. Suppose f is a smooth real-valued function on M, and consider the associated section f of the sheaf of germs of smooth functions on M. Explain (with examples) the relation between the set where f(m)=0 (a closed subset of M) and the set where fm=0 (an open subset of M, by one of your homework problems).
5. Given a Riemannian manifold M, let L be the laplacian on p-forms on M. Given a p-form b, when is there a p-form a with L(L(a))=b ? Give necessary and sufficient conditions in terms of the Hodge cohomology HpHod of M.