Timeline for Is there a relationship between tensor (or form) bundles and iterated tangent/cotangent bundles on a manifold?
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| Jan 11, 2013 at 13:04 | comment | added | Robert Bryant | Well, actually, there don't need to be multiple copies of the base manifold; you can always take the fiber product. For example, the fiber product $TM\oplus \Lambda^2(TM)$ is not the product of $TM$ with $\Lambda^2(TM)$. It's the set of pairs $(v,V)$ such that there exists an $x\in M$ with $v\in T_xM$ and $V\in \Lambda^2(T_xM)$. This is a vector bundle over a single copy of $M$. | |
| Jan 10, 2013 at 16:31 | vote | accept | Kensmosis | ||
| Jan 10, 2013 at 16:31 | comment | added | Kensmosis | Thanks, Robert. I'm a bit weak with Category Theory, so it took me a little digging to understand the prolongation explanation but I think I get the gist and it makes sense to me. In retrospect this may have been a somewhat naive question anyway, because there would be one "copy" of $M$ on the left side of my proposed equation and multiple copies of $M$ on the other side (one from each term) -- so the two wouldn't even be two homotopic in general. It just would have been nice to have some sort of "algebra" of these bundles. | |
| Jan 9, 2013 at 15:29 | history | answered | Robert Bryant | CC BY-SA 3.0 |