Differential Geometry Homework #8 due 12/7
1. (Boothby, p.81,#1) Let F : N → M be a one-to-one immersion which is proper, i.e., the inverse image of any compact set is compact. Show that F is an imbedding and that its image is closed regular submanifold of M .
2. Let ι : N → M be a one-to-one immersion, X be a manifold, and f : X → M be a smooth map with f (X) ⊆ ι(N ).
(a) Show by an example that ι−1◦ f : X → N may fail to be continuous.
(b) If ι−1◦ f is continuous, prove that it is smooth.
3. Let M(m, n) be the set of real m × n matrices and M(m, n; k) be the set of all m × n matrices of rank k. This exercise is to show that M(m, n; k) is a regular submanifold of M(m, n) of dimension k(m + n − k).
(a) For every M0 ∈ M(m, n; k) there exist permutation matrices P and Q such that
P M0Q =A0 B0 C0 D0
, where A0 is k × k non-singular matrix.
(b) There is some ε > 0 such that A is non-singular whenever all entries of A − A0 are < ε.
(c) If
A B
C D
where the entries of A − A0 are < ε, then X has rank k if and only if D = CA−1B.
(d) M(m, n; k) is a regular submanifold of M(m, n) of dimension k(m+n−k) for all k ≤ m, n.
1
