Exercise 1 This is a list of pathological manifolds.

Paracompact, 2nd count able, but not Hausdorff : a line with two origins.

Hausdorff, paracompact, but not second countable: un countable disjoint union of lines.

Hausdorff, not paracompact, not second countable: Prüfer surface (see Wiki or Spivak appen $\operatorname{dix}$ A) and the Long Line (see Wiki or Kobayashi and Nomizu page 166)

From now on, all the manifold are assumed to be Hausdorff, paracompact and second countable. Let us remark that $\mathbb{R}^m$ is locally compact, so every manifold is locally compact.

Exercise 4 This exercise shows that the pull-back is functorial (i.e. $(G \circ F)^=F^ G^$ ). The situation is the following $$M \stackrel{F}{\rightarrow} N \stackrel{G}{\rightarrow} N^{\prime} \stackrel{f}{\rightarrow} \mathbb{R}$$ We can pull-back the function $f$, and we have $$(G \circ F)^ f=F^* G^* f$$
To prove the functoriality of the pull-back of one forms there are two possibilities. A first proof is a computation in local co-ordinates. Let us first prove that $F^* d f=d F^* f$. Call $x=$ $\left(x_1, \cdots, x_N\right)$ and $y=\left(y_1 \cdots, y_n\right)$ the co-or dinates.
$$\begin{gathered} d\left(F^* f\right)=\sum_{i, k}\left(\partial_{x_i} f\right)\left(\partial_{y_k} F_i\right) d y_k \ F^*(d f)=\sum_i \partial_{x_i} f d F_i=\sum_{i, k}\left(\partial_{x_i} f\right)\left(\partial_{y_k} F_i\right) d y_k \end{gathered}$$

The other assertions follows from the previous computations working locally.
There second strategy is more intrinsic. A tangent vector $v$ is a derivation, its push-forward is defined as $\left(F_* v\right)(f):=v\left(F^* f\right)$. The functoriality of the push-forward follows from the functoriality of the pull back of functions. A 1 form $\omega$ is a linear form on the tangent vectors, its pull-back is defined as $\left(F^* \omega\right)(v)=\omega\left(F_* v\right)$, so the functoriality of the pull-back for one forms follows from the functoriality of the push-forward.

E-mail: help-assignment@gmail.com  微信:shuxuejun

help-assignment™是一个服务全球中国留学生的专业代写公司