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以下是一些我们可以帮助您解决的问题：
流形：流形的定义、性质和分类，如黎曼流形、流形的维度等。
度量和曲率：黎曼度量的定义和性质，曲率的计算和测量方法，以及曲率与度量之间的关系。
测地线：测地线的概念、性质和方程，测地线在黎曼流形中的重要性和应用。
切空间和余切空间：切空间和余切空间的定义和性质，切向量和余切向量的作用和计算。
黎曼几何定理：黎曼流形上的重要定理和结论，如黎曼流形的光锥性质、黎曼流形的共形变换等。
黎曼几何的应用：黎曼几何在物理学、计算机科学、数学物理等领域的实际应用。
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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.
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