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What is the meaning of Cotangent bundle tensor product tangent bundle:

$T^*M\otimes TM$? what will an element of this space be?

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This is isomorphic to $Hom(TM,TM)$. Namely it is the endomorphism bundle of the tangent bundle. In general given two vector spaces(or bundles) we have an isomorphism between $W \otimes V^*$ and $Hom(V,W)$.

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    $\begingroup$ You need $V$ to be finite-dimensional here. $\endgroup$ – Qiaochu Yuan Jan 20 '16 at 7:28
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    $\begingroup$ @QiaochuYuan Or for $W$ to be finite, no? Either way every element of $\mathrm{Hom}(V,W)$ has finite rank. $\endgroup$ – Oscar Cunningham Jan 21 '16 at 11:29

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