当使用这种包组合时，我的盒子里出现了一些奇怪的段落插入和删除现象。
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\begin{example}[Tangent Bundles]
Given a smooth manifold $M^n$, the \textit{tangent bundle} $TM$ is the union of all the tangent spaces of $M$, which has a natural topology and is also a smooth manifold. If $TM$ is trivial, i.e, splits as $TM \cong M \times \mathbb{R}^n$, then $M$ is called \textit{parallelizable}. For example, $S^1$ is parallelizable, but $S^2$ is not.
In fact, $TS^2$ does not admit even one non-vanishing section (i.e, a non-vanishing vector field); this is the hairy ball theorem which one can see e.g via Poincaré-Hopf (the sum of the degrees of the zeroes of a vector field on an orientable smooth manifold equal its Euler characteristic, and $\chi(S^2) = 2$).
\end{example}
\begin{example}[Tautological Bundles]
Consider $\mathbb{R} \mathbb{P}^n$ or $\mathbb{C} \mathbb{P}^n$ which parameterize lines in $\mathbb{R}^{n+1}$ and $\mathbb{C}^{n+1}$ respectively; the idea of the \textit{tautological bundle} on a projective space is that the line (real or complex) above a point is the line corresponding to that point. The total space is $\{([l],x) \in \mathbb{R}\mathbb{P}^n \times \mathbb{R}^{n+1}: x \in [l]\}$ equipped with the obvious projection maps (and similarly for $\mathbb{C}$). Simlilarly, one can define tautological bundles over Grassmanians $\mathrm{Gr}_k(\mathbb{R}^n)$ as above.
\end{example}
\begin{example}[Subbundles]
There is a natural notion of a subbundle of a vector bundle, whose definition we will not belabor here. As an example, take the trivial bundle $S^1 \times \mathbb{R}^2$, which has as a subbundle the Möbius strip (bundle), which one can see by picking a line through the origin, and then allowing it to rotate through a full circle as the base point varies over $S^1$ (this is exactly the construction of the tautological line bundle on $\mathbb{R} \mathbb{P}^1 = S^1$).
\end{example}
\begin{example}[Normal Bundles]
Given an embedding $M \hookrightarrow N$, we can consider its \textit{normal bundle} inside $N$, which I tend to think of interchangeably with a tubular neighborhood of the embedded copy of $M$ (technically we need a Riemannian structure on $N$ to take orthogonal complements). So, for example, if you know all of the possible vector bundles over $M$, then you know all of the possible tubular neighborhoods it can have in another manifold; I give an example of the utility of this below.
\end{example}
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产生这个：
这里有几个问题：首先是顶部的错位，这相当小。更严重的是删除了示例 0.1 中的段落分隔符，并插入了示例 0.4 中我没有放置的段落分隔符。我知道自动平衡 multicol 中的分隔 tcolorboxes 不会完美运行，但我以为它只是平衡性不好，而不是真正主动扰乱我的格式。还有其他事情发生吗，还是这是不可避免的？