Is the tensor product of a vector bundle with its dual bundle a trivial bundle?

Question

I wonder if it's true, if it's true, is there a easy to see why it is true?

Answer 1

The tensor product $E\otimes E^*$ of a vector bundle $E$ with its dual bundle $E^*$ is called the endomorphism bundle $\mathrm{End}(E)$. If you click $\rightarrow$ here $\leftarrow$ there's an example of such a non trivial real endomorphism bundle.

Answer : no it's not true that "the tensor product of a vector bundle with its dual bundle is a trivial bundle".

Remarks :

• Even though $\mathrm{End}(E)$ is generically non-trivial, $\mathrm{End}(E)$ admits a non-vanishing global section $\lambda$ given at each point $x$ of the base manifold $M$ by $\lambda|_x = \mathrm{id}_{E_x}$.
• There are special cases where $\mathrm {End}(E)$ is necessarily trivial. For example, if $E$ is a complex line bundle $L$, then the first Chern Class of $L\otimes L^*$ vanish and so $L\otimes L^*$ is trivial. Or again, since $L\otimes L^*$ has 1-dimensional fibers and has a non-vanishing global section then it is trivial.