Abstract. It is proved that every holomorphically convex complex space endowed with an action of a compact Lie group K can be realized as an open K-stable subspace of a holomorphically convex space endowed with a holomorphic action of the complexified group K . Similar results are obtained for holomorphic if-bundles over such spaces.Let G be a real Lie group which acts by holomorphic transformations on a (reduced) complex space X. Suppose that the Lie algebra of the complexification G c of G (see [Ho, p. 204]) is the complexification of the Lie algebra of G. This holds for example in the case where G is simply connected. Then, by integrating the holomorphic vector fields given by the G-action, the complexification G c acts locally and holomorphically on X (see [K]).Adapting the terminology of Palais (see [P]), we say that a complex space X* which contains X as an open subset is a globalization of the complex G-space X whenever the local G € -action on X extends to a global holomorphic action on X* and G c X = X*. The following results are proved in this paper.

On any real semisimple Lie group we consider a one-parameter family of left-invariant naturally reductive metrics. Their geodesic flow in terms of Killing curves, the Levi Civita connection and the main curvature properties are explicitly computed. Furthermore we present a group theoretical revisitation of a classical realization of all simply connected 3-dimensional manifolds with a transitive group of isometries due to L. Bianchi andÉ. Cartan. As a consequence one obtains a characterization of all naturally reductive left-invariant Riemannian metrics of SL(2, R).

We describe a relationship between globalizations of local holomorphic actions on Stein manifolds induced by global actions of certain non-compact Lie groups, and holomorphic fiber bundles with smooth Stein base and fiber and connected structure group. To this end we prove a univalence result for particular Stein Riemann domains with a free and properly discontinuous action of a discrete group of biholomorphisms. We then derive some consequences on the existence of Stein globalizations. Subject Classification (1991): 32M05, 32L05, 32D10, 32E10. Mathematics

Let G/K be a noncompact, rank-one, Riemannian symmetric space, and let G ‫ރ‬ be the universal complexification of G. We prove that a holomorphically separable, G-equivariant Riemann domain over G ‫ރ‬ /K ‫ރ‬ is necessarily univalent, provided that G is not a covering of SL(2, ‫.)ޒ‬ As a consequence, one obtains a univalence result for holomorphically separable, G×K -equivariant Riemann domains over G ‫ރ‬ . Here G×K acts on G ‫ރ‬ by left and right translations. The proof of such results involves a detailed study of the G-invariant complex geometry of the quotient G ‫ރ‬ /K ‫ރ‬ , including a complete classification of all its Stein G-invariant subdomains.

We carry out a detailed study of + , a distinguished G-invariant Stein domain in the complexification of an irreducible Hermitian symmetric space G/K . The domain + contains the crown domain and is naturally diffeomorphic to the anti-holomorphic tangent bundle of G/K . The unipotent parametrization of + introduced in Krötz and Opdam (GAFA Geom Funct Anal 18:1326-1421, 2008 and Krötz (Invent Math 172:277-288, 2008) suggests that + also admits the structure of a twisted bundle G × K N + , with fiber a nilpotent cone N + . Here we give a complete proof of this fact and use it to describe the G-orbit structure of + via the K -orbit structure of N + . In the tube case, we also single out a Stein, G-invariant domain contained in + \ which is relevant in the classification of envelopes of holomorphy of invariant subdomains of + .