Stein manifolds with compact symmetric center

The geometry of Grauert tubes and complexification of symmetric spaces

“…Now, for symmetric (or locally symmetric) spaces of nonnegative curvature the canonical complex structure exists on the whole tangent bundle T M (see for instance [22]). The resulting complex manifolds for compact rank one spaces are described by G. Patrizio and P-M. Wong in [19]. As a particular case, the Grauert tube complex structure on T S 2 associated to the standard round metric on S 2 is biholomorphic to the complex quadric {z For Riemannian manifolds with some negative sectional curvature, the whole tangent bundle cannot be given the structure of a Grauert domain.…”

Finsler metrics of constant curvature and the characterization of tube domains

“…It is known that (see [GS], [L], [LS], [PW1], [S]) given a complete Riemannian manifold M of nonnegative sectional curvature it is possible to define a complex structure on its tangent bundle T M so that M sits in T M as a totally real submanifold of top dimension. Moreover it is possible to define a smooth (in fact real analytic) function ∞ and satisfies on (1.1) and (1.2).…”

“…The first, which will be performed in Section 2, is to characterize flat simply connected Finsler manifolds. The second is to make the necessary adjustment in the arguments given in [PW1] for the case of Riemannian manifolds to show that T M is biholomorphic to C n and T (M ) to a tube domain. We shall outline this part in Section 3.…”

“…We shall outline this part in Section 3. Section 2 uses heavily results of [AP], and Section 3 of [PW1]; hence we shall rely freely upon these two references for notations, notions and technicalities, giving here only the elements needed to reconstruct the proofs.…”

Totally real orbits in affine quotients of reductive groups

“…In [1] there is a similar result if L contains a maximal torus of K. The main group theoretic ingredient there was the characterization of K/L as the unique totally c c real Jί-orbit in K /L . On the other hand, Patrizio and Wong construct in [9] special exhaustion functions on the complexification of symmetric spaces K/L of rank 1 and find that the minimum value of their exhaustions is always achieved on K/L. By a lemma of Harvey and Wells [6] one knows that the set where a strictly plurisubharmonic (briefly s.p.s.h) function achieves its minimum is totally real.…”