Normal bundle of a formal immersion

Let $(f,F)$ be a formal immersion from $M^m$ to $N^n$. The normal bundle of the formal immersion is the quotient vector bundle ${\nu }_F:=f^{\ast }TN/F(TM)$ over $M$, of rank $n-m$ when $m\le n$, where $f^{\ast }TN=M{\times }_{f}TN$ is the pullback bundle and $F(TM)\subseteq f^{\ast }TN$ is the image subbundle (a smooth subbundle because $F$ is a fibrewise injective bundle map over $f$). If $M=\mathrm{\varnothing }$ and $m>n$, the quotient is the empty bundle, which we regard as rank zero; no negative rank is assigned. If a smooth bundle metric is chosen on $f^{\ast }TN$, the fibrewise orthogonal complement of $F(TM)$ is a smooth subbundle of $f^{\ast }TN$ that represents ${\nu }_F$ canonically and identifies it with the orthogonal normal bundle; different metrics give canonically isomorphic orthogonal complements, so the isomorphism class of ${\nu }_F$ is intrinsic. When $(f,F)=(f,df)$ for a genuine immersion, ${\nu }_{df}$ is the normal bundle of the immersion, whose isomorphism class agrees with the normal bundle of the embedded image when the immersion is an embedding.