Proposition: The tangent-defect bundle

The quotient

has dimension two. It is identified with ${\mathbf{n}}^{\perp }$ by

Locally over $N$, these quotients form a smooth rank-two bundle. They are not established clean-composition fibers.

Proof. A vector belongs to ${{\mathcal{T}}_{\mathrm{lin}}}_w$ exactly when

The third cone equation follows from the copied linear relation.

The last two equations allow the decompositions

The tangent vectors to $N$ are precisely those with

for one $\delta \mathbf{n}\in {\mathbf{n}}^{\perp }$. Thus

The map is onto: for any $d_{\perp }\in {\mathbf{n}}^{\perp }$, choose $v=d_{\perp }/\alpha$ and $w=0$.

Its coefficients vary smoothly along $N$, its rank is two, and both ${\mathcal{T}}_{\mathrm{lin}}$ and $TN$ have constant ranks. The quotient is consequently a smooth local bundle.