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 section $s$ is transverse to $s_0(M)$ in the sense of [F6], so [F5] makes $Z=s^{-1}(s_0(M))$ an embedded submanifold of $M$ of codimension $r$; it is closed because $s_0(M)$ is closed and $s$ is continuous, so $\dim Z=n-r$ when nonempty. 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. 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 ${\mathbb{F}}_2$ both sides carry their unique nonzero generator.