Orthogonal complement of a vector bundle

Question

Let $E \rightarrow X$ be a vector bundle with an inner product. If $F$ is a sub-bundle, we can define an orthogonal complement bundle $F^\perp$ (see http://www.math.cornell.edu/~hatcher/VBKT/VB.pdf for the construction, and the source of the problem). I am trying to show that $F^\perp$ is independent of the choice of inner product, up to isomorphism. I have tried to do a local construction, defining the isomorphism in each locally trivial neighborhood, but I do not see a way to patch these together into a global isomorphism. Any suggestions?

Answer 1

Hint: First show that for an inner product space $(V, \langle \cdot, \cdot \rangle)$ and a subspace $W \subseteq V$, $$W^\perp \cong V/W.$$ Then, using the quotient bundle construction of the previous exercise (exercise 2), show that for a Euclidean vector bundle $E \longrightarrow X$ with sub-bundle $F$, $$F^\perp \cong E/F.$$ Since $E/F$ is independent of the bundle metric, this will show that $F^\perp$ does not depend on the bundle metric (up to isomorphism).

Let me know if you want me to provide the details of the argument.