# When are the vector bundles $E$ and $E^\vee \otimes \det(E)$ isomorphic?

Let $$C$$ be a smooth complex projective curve and let $$E$$ be a rank two vector bundle over $$C$$. If $$E$$ is decomposable, ie $$E=L\oplus M$$ for some line bundles $$L$$ and $$M$$ we have that $$E^\vee \otimes \det(E) = (L^\vee\oplus M^\vee)\otimes L\otimes M = M\oplus L \simeq E.$$ My question is: Under which conditions does it happen to an indecomposable $$E$$?

More generally, If $$E$$ is an indecomposable vector budle of arbitrary rank over a compact complex manifold $$X$$. Under which conditions $$E\simeq E^\vee \otimes L$$ for some line bundle $$L$$?

For any rank 2 vector bundle there is a pairing $$E \otimes E \to \det(E).$$ Being non-degenerate it induces an isomorphism $$E \cong E^\vee \otimes \det(E)$$, which thus holds for any rank 2 vector bundle.
For the more general question --- such an isomorphism exists if and only if there is a non-degenerate pairing $$E \otimes E \to L$$.