Kaehler structures on

Let be a compact connected semi-simple Lie group, let , and let be an Iwasawa decomposition. To a given -invariant Kaehler structure on , there corresponds a pre-quantum line bundle on . Following a suggestion of A.S. Schwarz, in a joint paper with V. Guillemin, we studied its holomorphic sections as a -representation space. We defined a -invariant -structure on , and let denote the space of square-integrable holomorphic sections. Then is a unitary -representation space, but not all unitary irreducible -representations occur as subrepresentations of . This paper serves as a continuation of that work, by generalizing the space considered. Let be a Borel subgroup containing , with commutator subgroup . Instead of working with , we consider , for all parabolic subgroups containing . We carry out a similar construction, and recover in the unitary irreducible -representations previously missing. As a result, we use these holomorphic sections to construct a model for : a unitary -representation in which every irreducible -representation occurs with multiplicity one.

     

Generic warped product submanifolds of locally conformal keahler manifolds

Warped product manifolds are known to have applications in physics. For instance, they provide an excellent setting to model space-time near a black hole or a massive star (cf. [9]). The studies on warped product manifolds with extrinsic geometric point of view were intensified after the B.Y. Chen's work on CR-warped product submanifolds of Kaehler manifolds (cf. [6], [7]). Later on, similar studies were carried out in the setting of l.c.K. manifolds and nearly Kaehler manifolds (cf. [3], [11]). In the present article, we investigate a larger class of warped product submanifolds of l.c.K. manifolds, ensure their existence by constructing an example of such manifolds and obtain some important properties of these submanifolds. With regard to the CR-warped product submanifold, a special case of generic warped product submanifolds, we obtain a characterization under which a CR-submanifold is reducesd to a CR-warped product submanifold.     

复空间形式中曲率齐性超曲面

本文证明了复空间形式中曲率齐性kaehler超曲面是全测地的或局部全纯等距于复射影空间cpn+1（c）（c>0）的超二次曲面Qn,还讨论了cp2（1）中曲率齐性实超曲面。     

Harmonic Riemannian maps on locally conformal Kaehler manifolds

We study harmonic Riemannian maps on locally conformal Kaehler manifolds (lcK manifolds). We show that if a Riemannian holomorphic map between lcK manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, we prove that a Riemannian holomorphic map is harmonic if and only if the lcK manifold is Kaehler. Then we find similar results for Riemannian maps between lcK manifolds and Sasakian manifolds. Finally, we check the constancy of some maps between almost complex (or almost contact) manifolds and almost product manifolds.     

GEOMETRIC INEQUALITIES FOR CERTAIN SUBMANIFOLDS IN A PINCHED RIEMANNIAN MANIFOLD

This article gives some geometric inequalities for a submanifold with parallel second fundamental form in a pinched Riemannian manifold and the distribution for the square norm of its second fundamental form.     

ON A CONJECTURE OF K. OGIUE FOR KAEHLER HYPERSURFACES

ＯＮ Ａ ＣＯＮＪＥＣＴＵＲＥ ＯＦ Ｋ． ＯＧＩＵＥ ＦＯＲ ＫＡＥＨＬＥＲ ＨＹＰＥＲＳＵＲＦＡＣＥＳ￥ＳＨＡＮＧＷＥＩＭＩＮＧＡｂｓｔｒａｃｔ：ＡｎａｆｆｉｒｍａｔｉｖｅａｎｓｗｅｒｔｏａｃｏｎｊｅｃｔｕｒｅｏｆＫ．Ｏｇｉｕｅｆｏｒｍｕｌａｔｅ...     

Complete Kaehler Submanifolds in CP~n

In this paper we mainly investigate projectively flat complete Kaehler sub-manifolds, in CPn. We give the pinching constants and the local structure.     

Vanishing theorems on Riemannian manifolds, and geometric applications

A general Liouville-type result and a corresponding vanishing theorem are proved under minimal regularity assumptions. The latter is then applied to conformal deformations of stable minimal hypersurfaces, to the L² cohomology of complete manifolds, to harmonic maps under various geometric assumptions, and to the topology of submanifolds of Cartan-Hadamard spaces with controlled extrinsic geometry.     

ON FACTORIZATION THEOREMS OF PLURIHARMONIC MAPS INTO THE UNITARY GROUP

ＯＮＦＡＣＴＯＲＩＺＡＴＩＯＮＴＨＥＯＲＥＭＳＯＦＰＬＵＲＩＨＡＲＭＯＮＩＣＭＡＰＳＩＮＴＯＴＨＥＵＮＩＴＡＲＹＧＲＯＵＰＣＨＥＮＧＱＩＹＵＡＮＤＯＮＧＹＵＸＩＮａｂｓｔｒａｃｔＴｈｅａｕｔｈｏｒｓｇｉｖｅｓｏｍｅｃｏｎｓｔｒｕｃｔｉｖｅ...     

Non-existence of Warped Product Semi-Slant Submanifolds of Kaehler Manifolds

In this paper, we show that there are no warped product semi-slant submanifolds of Kaehler manifolds. Contrary to this result,we provide an elementary example of a CR-warped product submanifold of a Kaehler manifold