So I have a problem that I encountered and I don't know what object I am looking at. So there are two manifolds $M_1$ and $M_2$ and a smooth map $f:M_1\rightarrow M_2$. We know that there is a vector bundle $\pi_2:E_2\rightarrow M_2$, and we also have a pullback vector bundle $\pi_1:f^\ast E_2\rightarrow M_1$. I want to know what exactly is the vector bundle with total space $T^\ast M_1\otimes f^\ast(E_2)$. I'm supposed to find a smooth section in this vector bundle, but I have no idea what the vector bundle is. Is it the map $T^\ast E_1\otimes f^\ast(E_2)\rightarrow M_1\otimes M_1$, this construction didn't make sense to me, so I thought it might be $T^\ast M_1\otimes f^\ast(E_2)\rightarrow T^\ast M_1$. Any thoughts on what it should be?
For reference this is the problem I'm looking at. It is part (c)
Best Answer
Given two vector bundles $V_1$, $V_2$ over $M$, we can form their tensor product bundle $V_1\otimes V_2$ which is a vector bundle over $M$. The fibre of the tensor product bundle is the tensor product of the fibres of the original two bundles. For more details on the construction of the tensor product bundle, see page $13$ of Hatcher's Vector Bundles and K-Theory.
In your question $T^*M_1$ and $f^*E_2$ are both vector bundles over $M_1$, so $T^*M_1\otimes f^*E_2$ is a vector bundle over $M_1$.