Tensorproduct of vectorbundles

Question

Assume we got $\pi:E \to M$, a vector bundle of manifolds. I know how to make the bundle $(E \otimes E)^* \to M$. But how do the local trivializations look like ? I suppose that if $(\varphi_\alpha,U_\alpha)$ are local trivializations of $\pi$ then those of $(E \otimes E)^*$ are given by $\phi_\alpha: (\mathbb R^n \otimes \mathbb R^n)^* \times U_\alpha \to (E\otimes E)^*_{U_\alpha}$ via $$ \phi_\alpha(\eta,x)(u \otimes v) = \eta(\pi_1(\varphi_{U_\alpha}^{-1}(u)) \otimes \pi_1(\varphi^{-1}_{U_\alpha}(v))) $$

Answer 1

A remark: vector bundle of manifolds is a misleading name, you probably want to say a vector bundle $E$ over a manifold $M$.

I shall assume that the fibres of $E$ are isomorphic to $\mathbb{R}^k$. From the trivialisations $\varphi_\alpha:E\big{|}_{U_\alpha}\to\mathbb{R}^k\times U_\alpha$, one can define $\varphi^*_\alpha:E^*\big{|}_{U_\alpha}\to(\mathbb{R}^k)^*\times U_\alpha$ by $\varphi^*_\alpha\big{|}_p(f)=f\circ(\varphi_\alpha\big{|}_p)^{-1}$, where $p\in M$, $E_p$ is the fibre of $E$ over the point $p$, $f:E_p\to\mathbb{R}$ belongs to $E_p^*$, and $\varphi_\alpha\big{|}_p:E_p\to\mathbb{R}^k\times \{p\}$ is the isomorphism between the fibre $E_p$ and $\mathbb{R}^k$ provided by the trivialisation $\varphi_\alpha$ (I am identifying $\mathbb{R}^k\times \{p\}$ and $\mathbb{R}^k$).

Now it is just a matter of taking the tensor product: $\phi_\alpha:(E\otimes E^*)\big{|}_{U_\alpha}\to\mathbb{R}^k\otimes(\mathbb{R}^k)^*\times U_\alpha$ can be defined by $\phi_\alpha\big{|}_p(v\otimes f)=\varphi_\alpha\big{|}_p(v)\otimes f\circ(\varphi_\alpha\big{|}_p)^{-1}$, with $v\in E$ and $f\in E^*$.