Unramified Riemann Domains Satisfying the Oka–Grauert Principle over a Stein Manifold

Let \((D, \pi )\) $(D,\pi )$ be an unramified Riemann domain over a Stein manifold of dimension n. Assume that \(H^k(D,\mathscr {O}) = 0\) $H^k(D,\mathcal{O})=0$ for \(2 \le k \le n - 1\) $2\le k\le n-1$ and there exists a complex Lie group G of positive dimension such that all differentiably trivial holomorphic principal G-bundles on D are holomorphically trivial. Then, we prove that D is Stein.