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If I have correctly undrestood,it is a result of the so called Grauert-Oka principle that all holomorphic vector bundles over contractible Stein manifolds are holomorhically trivial.Does any one knows about a more direct proof?

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    $\begingroup$ Oka-Grauert tells you that the classification of holomorphic and topological vector bundles coincide. But being simply connected isn't sufficient for topological vector bundles to be trivial. $\endgroup$
    – Qiaochu Yuan
    Commented Feb 18, 2015 at 20:20
  • $\begingroup$ Yuan, I changed the question a liitle bit. $\endgroup$
    – Hamed
    Commented Feb 18, 2015 at 20:27

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This is a theorem which is known in classical Oka-Grauert theory , You can find the direct proof in following book page 220 and 190

Stein Manifolds and Holomorphic Mappings: The Homotopy Principle in Complex ... by Franc Forstneri

Here is google book Link

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