Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2015
Math 636 — Problem Set 9
Issued: 11.08
Due: 11.16
9.1. Liouville numbers. A real number z is called Liouville
every
if for
p
positive integer there exist integers p and q such that z − q < q1n .
Prove that the set of Liouville numbers is comeagre (residual) and has
Lebesgue measure 0.
9.2. Projective space. (a) Prove that the coordinate charts
xi−1 xi+1
xn
x0
,...,
,
,...,
Pi : [x0 , x1 , . . . , xn ] 7→
xi
xi
xi
xi
on the projective space RP n are C ∞ -related.
(b) Prove that the manifold RP 1 and the circle (with the natural differentiable structure) are diffeomorphic.
9.3. Frames and Grassman manifolds. (a) Show that the set F(m, k)
of all linearly independent k-tuples of vectors in Rn is a differentiable
manifold.
(b) Show that the set G(k, n) of all k-dimensional subspaces of the
vector space Rn is a differentiable manifold. (Note that G(1, n) = RP n .)