Normal bundle of the zero locus of a transverse section

The zero section $s_0:M\to E$ is a smooth embedding [F1], so by [F2] its normal bundle is the quotient ${\nu }_{s_0}=TE{\mathrm{\mid }}_{s_0(M)}\mathrm{/}T(s_0(M))$. The canonical splitting $TE{\mathrm{\mid }}_{s_0(M)}\cong TM\oplus E$ along the zero section, whose vertical summand is the fibre direction, identifies this quotient with $E$ as smooth bundles over $M$ by [F3] and [F4]; moreover $s_0(M)$ is closed in $E$ because in every bundle chart its complement is the open set of nonzero vectors.

The zero section has codimension $r$ by the bundle coordinates in step 1.1. Since $s$ is transverse to it by [F6], the $r>n$ branch of [F5] gives $Z=\mathrm{\varnothing }$ in that range. Its two normal-bundle sets are then empty: the normal quotient is the empty coproduct and $E{\mathrm{\mid }}_Z$ is the empty pullback. By [F10] their unique map is a smooth bundle isomorphism with the specified rank-$r$ empty-bundle model; use the empty zero-dimensional structure on $Z$ without asserting codimension $r$. The same empty-map conclusion holds if $Z=\mathrm{\varnothing }$ for $r\le n$, now with its allowed dimension $n-r$. In either case $Z$ is closed and all pointwise tangent and orientation statements are vacuous. For $r=0$, the bundle has one zero vector in every fibre, so $s=s_0$, $Z=M$ and the normal identification is the canonical map between zero bundles. It remains to handle $1\le r\le n$ at actual zeros.

In this dimension range, [F5] makes $Z=s^{-1}(s_0(M))$ an embedded submanifold of codimension $r$; it is closed because $s_0(M)$ is closed and $s$ is continuous, so $\dim Z=n-r$. The quotient map $D^{v}s:TM{\mathrm{\mid }}_Z\to E{\mathrm{\mid }}_Z$ has kernel $TZ$ by [F5] and is surjective by transversality. In bundle charts it is the derivative of the local section components at their zeros, hence smooth; it induces a smooth fibrewise isomorphism $TM{\mathrm{\mid }}_Z\mathrm{/}TZ\to E{\mathrm{\mid }}_Z$, whose inverse is smooth by the inverse-matrix formula. Here $E{\mathrm{\mid }}_Z$ is the pullback along the inclusion $Z\hookrightarrow M$, not along the section $s:M\to E$. Combining with step 1.1 gives the canonical isomorphism ${\nu }_Z\cong E{\mathrm{\mid }}_Z$ of smooth bundles over $Z$.

Orientation clause. If $Z$ is empty, the orientation data form the unique empty assignment, compatible with step 2.1. Otherwise the isomorphism of step 1.1 carries the orientation of $E$ to the normal orientation of $s_0(M)$ induced by the ambient $E$ and the zero-section orientation, by the third-orientation rule [F9] applied with the total-space orientation in which a positive tangent basis of $s_0(M)$ followed by a positive fibre basis is positive (the tangent-first convention of this pair). Pullback along $s$ preserves this ordered determinant-line comparison because $ds$ maps the normal directions of the transverse preimage isomorphically onto the normal directions of $s_0(M)$ by [F5] and [F8], and the induced orientation of $Z$ in $M$ is the one for which a positive tangent basis of $Z$ followed by a positive normal basis is positive in $M$ [F9]. Hence the orientation of ${\nu }_Z$ induced from $M$ and $Z$ corresponds to the supplied fibre orientation of $E{\mathrm{\mid }}_Z$; over ${\mathbb{F}}_2$ the fibre orientation data at each actual point are canonical. No nonzero global generator is asserted over the empty base.