MATH 625 Homework

Assignment #6, Due 10 Oct

Write your name on each assignment. Homework is due at the beginning of class.

THERE WILL BE NO CREDIT FOR LATE HOMEWORK.

1. Let \(M\) be \(\mathbb{R}^2\) with the standard smooth manifold structure, which is determined by the \(\mathbb{R}^2\)-valued atlas \(\mathcal{A}_M = \{(\mathbb{R}^2,x)\}\), where \(x(p_1,p_2) = (p_1,p_2)\) is the identity map on \(\mathbb{R}^2\).

   Let \(N\) be \(\mathbb{R}^2\) with the smooth structure that is determined by the \(\mathbb{R}^1\)-valued atlas \(\mathcal{A}_N = \{(\mathbb{R}\times\{\beta\},y_{\beta})\}_{\beta\in\mathbb{R}}\), where \(y_{\beta}(p,\beta)=p\). Observe that \(N\) is a \(1\)-dimensional manifold.

   Let \(f : M \rightarrow N\) be the map defined by \(f(p_1,p_2)=(p_1,p_2)\). As a map from \(\mathbb{R}^2\) to \(\mathbb{R}^2\),  \(f\) is the identity map.

   1. For each \(\beta \in \mathbb{R}\), show that \[ y_{\beta}\circ f\circ x^{-1} : x(\mathbb{R}^2 \cap f^{-1}(\mathbb{R}\times\{\beta\})) \rightarrow \mathbb{R} \] is a \(C^{\infty}\) map.
   2. Show that \(f : M \rightarrow N\) is not \(C^{\infty}\), and explain why that does not contradict Proposition 1.55 on page 22.
2. Let \(M=\mathbb{R}\mathrm{P}^n\) be \(n\)-dimensional real projective space with the standard smooth manifold structure (see Example 1.41 on page 18, or consult your class notes). Recall that a point in \(M\) is a line \(\ell\) through the origin in \(\mathbb{R}^{n+1}\). Let \(E\) be the subset of \(M\times\mathbb{R}^{n+1}\) defined by \[ E = \{ (\ell , v) \in M\times\mathbb{R}^{n+1} \mid v \in \ell\}, \] and define the surjective map \(\pi : E \rightarrow M\) by \(\pi(\ell,v) = \ell\).
   1. Fix \(\ell\in M\), and consider the pre-image \(\pi^{-1}(\{\ell\})\subset E\). Find a natural way to endow this pre-image with the structure of a 1-dimensional vector space. In other words, define addition (\((\ell,v) + (\ell,w)\)) and scalar multiplication (\(c\cdot(\ell,v)\)) for the points in \(\pi^{-1}(\{\ell\})\).
   2. Find an \(\mathbb{R}^{n+1}\)-valued smooth atlas for \(E\) such that \(\pi : E \rightarrow M\) is a smooth map. Remark: we are creating our first vector bundle.
3. Page 26: Exercise 1.61
4. Page 26: Exercise 1.63
5. Page 26: Exercise 1.64

Last modified: Tue Oct 4 11:53:58 HST 2016