Differential Geometry Homework #7 due 11/30
1. We define a k-frame in Rn to be a linear independent set x of k elements of Rn:
x1 = (x11, · · · , xn1) ...
xk = (x1k, · · · , xnk).
A k-frame in Rn may be identified with the k × n matrix, which we also denoted by x, whose rows are x1, · · · , xk. Let F (k, n) be the set of k-frame in Rn.
(a) Show that F (k, n) is differentiable manifold of dimension kn.
(b) Let x and y be two k-frames. We define an equivalence relation ∼ by saying that
x ∼ y if y = ax, a ∈ GL(k, R).
Then G(k, n) is identified with F (k, n)/ ∼. Let π : F (k, n) → G(k, n) be the quotient map. Show that π is in fact C∞.
2. (Boothby p.74 #3) On the ”figure eight” image ˜N of R on page 72.
Let the topology and C∞ structure be given by the one-to-one immersion G (please refer to page 72 for the definition of G). Show that reflection H : (x1, x2) → (x1, −x2) of R2 in the x1 axis, although it maps ˜N onto itself, is not a diffeomorphism of ˜N .
3. Let M be a compact manifold of dimension n and F : M → Rn be a smooth map. Prove that F cannot everywhere be non-singular.
4. Consider S1 as the unit circle in the complex plane. Define a mapping F : R → S1 × S1 by setting F (t) = (e2πit, e2πiαt), where α is an irrational number. Prove that F is an injective immersion with dense range.
5. Let Φ : R2 → R be defined by
Φ(x, y) = x3+ xy + y3+ 1.
Which level sets of Φ are regular submanifolds of R2.
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