Good point, I have changed “of” to “on”.

]]>Also, it’s not an inner product *of* vector bundles, rather *on* vector bundles. I was expecting some sort of categorified thing $Vect(X)\times Vect(X) \to$???

Woops, sorry. My mind was still on the entry on direct sum of vector bundles, it seems. Fixed now.

]]>Well, you definitely don’t want to say “vector bundle map $E \oplus_X E \to X \times \mathbb{R}$”, because that’s just wrong. I’d think it best to follow Dmitri’s advice, and then break it down into bilinear maps if that seems too high-falutin’.

]]>Wouldn’t it be better to simply use ⊗?

]]>I think we should change $E\oplus_X E$ to $E\times_X E$ and $E_x\oplus E_x$ to $E_x\times E_x$. In my mind $V\oplus V$ is a vector space whereas $V\times V$ is a just set. Writing $E_x\oplus E_x\to\mathbb{R}$ makes me think that the map is linear, whereas $E_x\times E_x\to\mathbb{R}$ would just be a function. This is how it’s done on the page “inner product space”, it uses $\times$ rather than $\oplus$.

]]>created *inner product of vector bundles* with the construction over paracompact Hausdorff spaces