Stein extensions of real symmetric spaces and the geometry of the flag manifold

 Let G be a connected semisimple Lie group contained in its simply connected complexification G _C . Let KG∩K _C be a maximal compact subgroup of G. Denote by X _o the unique closed G-orbit in the full flag manifold ℱ and by 𝒪 the unique open K _C -orbit in ℱ. The set consisting of the elements gK _C so that gX _o ⊂𝒪 is an Stein extension of G/K⊂G _C /K _C . There is a universal domain , natural form the point of view of group actions which has been conjectured to be Stein. The main result of this paper is the inclusion . In the second part of the paper I show, under some dominance condition in the parameter, that representations in Dolbeault cohomology can be realized as holomorphic sections of vector bundles over . Received: 9 September 2002 / Revised version: 12 July 2002 / Published online: 8 April 2003 Mathematics Subject Classification (2002): 22E30 Research partially supported by NSF grant DMS-9801605 and DMS 0074991.     

Special isoparametric orbits in Riemannian symmetric spaces

In the present paper orbits of isotropy subgroups in Riemannian symmetric spaces are discussed. Principal orbits of an isotropy subgroup are isoparametric in the sense of Palais and Terng (seeCritical Point Theory and Submanifold Geometry, Springer-Verlag, Berlin, 1988). We show that excepting some special cases, the shape operator with respect to the radial unit vector field determines a totally geodesic foliation on a given principal orbit. Furthermore, we prove that the shape operators and the curvature endomorphisms with respect to the normal vectors commute on these isoparametric submanifolds.     

Asymptotic in Sobolev spaces for symmetric Paneitz-type equations on Riemannian manifolds

We describe the asymptotic behaviour in Sobolev spaces of sequences of solutions of Paneitz-type equations [Eq. (E _α ) below] on a compact Riemannian manifold (M, g) which are invariant by a subgroup of the group of isometries of (M, g). We also prove pointwise estimates.     

Riemannian flag manifolds with homogeneous geodesics

A geodesic in a Riemannian homogeneous manifold is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of the Lie group . We investigate -invariant metrics with homogeneous geodesics (i.e., such that all geodesics are homogeneous) when is a flag manifold, that is, an adjoint orbit of a compact semisimple Lie group . We use an important invariant of a flag manifold , its -root system, to give a simple necessary condition that admits a non-standard -invariant metric with homogeneous geodesics. Hence, the problem reduces substantially to the study of a short list of prospective flag manifolds. A common feature of these spaces is that their isotropy representation has two irreducible components. We prove that among all flag manifolds of a simple Lie group , only the manifold of complex structures in , and the complex projective space admit a non-naturally reductive invariant metric with homogeneous geodesics. In all other cases the only -invariant metric with homogeneous geodesics is the metric which is homothetic to the standard metric (i.e., the metric associated to the negative of the Killing form of the Lie algebra of ). According to F. Podestà and G.Thorbergsson (2003), these manifolds are the only non-Hermitian symmetric flag manifolds with coisotropic action of the stabilizer.

     

Riesz transform on locally symmetric spaces and Riemannian manifolds with a spectral gap

In this paper we study the Riesz transform on complete and connected Riemannian manifolds M with a certain spectral gap in the L² spectrum of the Laplacian. We show that on such manifolds the Riesz transform is Lp bounded for all p∈(1,∞). This generalizes a result by Mandouvalos and Marias and extends a result by Auscher, Coulhon, Duong, and Hofmann to the case where zero is an isolated point of the L² spectrum of the Laplacian.     

Compression semigroups of open orbits on real flag manifolds

Let (G, H) be an irreducible semisimple symmetric pair,P G a parabolic subgroup. Suppose that theL-orbit of the base point in the flag manifoldG/P is open and writeS(L,P)={gG:gL LP} for the compression semigroup of this orbit. We show that ifP is minimal andS(L, P)=G, then (G, H) is Riemannian and we give a geometric characterization of those cases whereS(L, P) has non-empty interior different fromG. IfG/H is a symmetric space of regular type, then we show under certain additional assumptions thatS(L, Q) is an Ol'shanskiî semigroup.Supported by a DFG Heisenberg-grant.     

Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces

In this paper we define a distinguished boundary for the complex crowns of non-compact Riemannian symmetric spaces . The basic result is that affine symmetric spaces of can appear as a component of this boundary if and only if they are non-compactly causal symmetric spaces.

     

Jordan triples and Riemannian symmetric spaces

We introduce a class of real Jordan triple systems, called JH-triples, and show, via the Tits-Kantor-Koecher construction of Lie algebras, that they correspond to a class of Riemannian symmetric spaces including the Hermitian symmetric spaces and the symmetric R-spaces.     

On locally symmetric vector fields on Riemannian manifolds

It is shown that if ann-dimensional (n≧3) Riemannian manifold admitsr≧2 locally symmetric vector fields (LSVF's), then it is aV(k)-space. In particular, ifr=n−1 then the manifold is a space of constant curvature. In the case of a 3-dimensional Riemannian manifold a close connection between LSVF's and eigenvectors of the Ricci tensor is found.     

Pseudo-Riemannian manifolds modelled on symmetric spaces

We construct irreducible pseudo-Riemannian manifolds (M, g) of arbitrary signature (p, q) with the same curvature tensor as a pseudo-Riemannian symmetric space which is a direct product of a two-dimensional Riemannian space form M ₂(c) and a pseudo-Euclidean space with the signature (p, q ? 2), or (p ? 2, q), respectively.     

Pseudo-Riemannian manifolds modelled on symmetric spaces

We construct irreducible pseudo-Riemannian manifolds (M, g) of arbitrary signature (p, q) with the same curvature tensor as a pseudo-Riemannian symmetric space which is a direct product of a two-dimensional Riemannian space form M ₂(c) and a pseudo-Euclidean space with the signature (p, q − 2), or (p − 2, q), respectively.     

Invariants,orbits and symmetric spaces

Lavoro svolto nell'ambito del G.N.S.A.G.A. del C.N.R. con contributo del M.P.I. fondi 40%.