Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 17 / 1 / b / Solution

Let $r=\mathrm{rank}F$ and $s=\mathrm{rank}E$. Around any $p$, choose a local frame $f_1,\dots ,f_r$ of the vector subbundle $F$. Choose additional smooth sections $e_{r+1},\dots ,e_s$ of $E$ which complement these vectors at $p$. They can be obtained by constant-coordinate sections in a vector bundle trivialization. The determinant of the resulting frame is nonzero at $p$, hence on a sufficiently small neighborhood. Thusis an adapted local frame of $E$ near $p$.

On the fiberwise quotient, the classes $[e_{r+1}],\dots ,[e_s]$ form a basis. Give the quotient vector bundle its local coordinates byAn overlap between adapted frames has a smooth transition function of a vector bundle represented by an invertible block upper triangular matrix,The zero lower-left block expresses that both sets of first $r$ frame vectors span $F$. Its lower-right block $D$ is smooth and invertible. Passing to quotient coordinates removes the upper block, leaving precisely the transition $D$. These lower-right blocks satisfy the cocycle identity because the original transition matrices do. Consequently they define a compatible smooth bundle atlas of rank $s-r$. This proves $E/F$ is a smooth vector bundle, and its fiberwise quotient map $q:E\to E/F$ is a smooth vector bundle morphism. The construction is independent of the adapted frames: any two choices have the same type of compatible transition. The usual quotient topology agrees with this atlas, since in adapted coordinates $q$ is an open linear projection on each fiber.