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Let $M$ be a smooth manifold. The tangent bundle is naturally a smooth vector bundle, but it obviously has more structure than that. Specifically, there is a natural action of the the diffeomorphism group of $M$ on $TM$. Unless I am mistaken, this action is what distinguishes $TM$ it from an isomorphic vector bundle (e.g., the cotangent bundle of $M$). A similar question could be asked for the bundle of $n$-forms and the bundle of densities on an orientable $n$-manifold. Both should be trivial line bundles, but the action should be different.

My question is what are good ways to think about this additional structure and does it have a name? The less category theory the better.

Rbega
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1 Answers1

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I think what you are looking for is the concept of a natural vector bundle. Basically, this means that this is a way how to associate a vector bundle to any manifold (of some fixed dimension) and of lift of any diffeomorphism and any open emdbedding between two manifolds of the same dimension to vector bundle homomorphisms with obvious compatibility conditions. (Fomally, it is a functor from manifolds and local diffeomorphisms to vector bundles which assigns to any $M$ a bundle over $M$, but that's the category theory that you want to avoid.) You can find a lot about this concept in the book by Kolar, Michor and Slovak, see here.

By the way: Over an oriented manifold, $n$-forms and densities are isomorphic (even as natural vector bundles). This is the reason why $n$-forms can be integrated on an oriented manifold.

Andreas Cap
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  • Thanks for the reference, I'll check it out. That being said, your last comment makes me think it is not quite what I am trying to understand. For instance on $S^2$ I would would expect the antipodal involution to pull back the standard (i.e. associated to the round metric) volume form to it's negative, while it should pull back the standard density to itself. I guess maybe the natural part restricts diffeomorphisms of oriented manifolds to the orientation preserving ones. – Rbega Sep 08 '17 at 20:13
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    I am sorry, the last part of my answer was not completely correct.Densities and $n$-forms are non-isomorphic as natural bundles over manifolds and local diffeomorphisms. They are isomorphic as natural vector bundles over oriented manifolds and orientation preserving local diffeomorphisms. (So the situation is still a bit different from the case of the tangent and cotangent bundle. I don't think that these are isomorphic as natural vector bundles on some reasonable category.) – Andreas Cap Sep 09 '17 at 09:01