= Defining line bundle of a properly embedded hypersurface
Let $Y\subseteq X$ be a properly embedded <hypersurface>. Choose local defining <smooth functions> $f_\alpha$ with nonzero <differentials> along $Y$. On overlaps they differ by nowhere-zero smooth factors. These transition factors define a <real line bundle> $L\to X$ and a <transverse section> $s$ whose zero set is $Y$.
The <derivative> of $s$ along its zero set is intrinsic and gives a surjective bundle map
$$
ds:TX|_Y\longrightarrow L|_Y.
$$
Its kernel is $TY$, so it induces the <vector bundle isomorphism>
$$
N_{Y/X}=TX|_Y/TY\cong L|_Y.
$$
Thus the restriction of the defining <line bundle> is the <normal bundle>. These bundles have different original base spaces: $L$ lives over $X$, while $N_{Y/X}$ lives over $Y$. A global real-valued regular defining <function> would trivialize that restriction, explaining the obstruction from a nontrivial <normal bundle>.
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