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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 26th 2017

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

  1. I think we should change E XEE\oplus_X E to E× XEE\times_X E and E xE xE_x\oplus E_x to E x×E xE_x\times E_x. In my mind VVV\oplus V is a vector space whereas V×VV\times V is a just set. Writing E xE xE_x\oplus E_x\to\mathbb{R} makes me think that the map is linear, whereas E x×E xE_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.

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 26th 2017

    Wouldn’t it be better to simply use ⊗?

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 26th 2017

    Well, you definitely don’t want to say “vector bundle map E XEX×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’.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 26th 2017

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

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 26th 2017

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

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 26th 2017

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

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