The article studies small neighbourhoods of real forms in complex Lie groups. These are the main results:
Let G be a complex Lie group, H a real form, p: G -> G/H the natural projection. Then every point of G/H admits a neighbourhood basis formed by open subsets U of G/H for which the preimage W=p-1(U) is complete hyperbolic and Stein.
Let G be a simply connected complex Lie group and H a real form of G such that the maximal semisimple Lie subgroup of H is compact. Let p: G-> G/H denote the natural projection. Then for every relatively compact open subset U of G/H the preimage W=p-1(U) is hyperbolic.
Let G be a connected real Lie group. Then there exists a complete hyperbolic Stein complex manifold X on which G acts freely and effectively.
manuscripta math. 79 , 329--334 (1993)