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Totally complex submanifolds of a quaternion projective space

四元数射影空间的全复子流形

基本信息

批准号：
15540065
负责人：
TSUKADA Kazumi
金额：
$ 1.66万
依托单位：
OCHANOMIZU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540065/
关键词：
a quaternion projective space a quaternionic Kahler manifold totally complex submanifolds twistor space totally real submanifolds Einstein-Kahler submanifolds a complex quadric isotropic Kahler immersions 複素二次超曲面複素共形構造ルジャンドル部分多様体

项目摘要

In this research, we investigate totally complex submanifolds of a quaternion projective space (more generally a quaternionic Kahler manifold). It is one aspect of the interplay of quaternionic differential geometry and complex differential geometry and makes so-called "quaternionic complex differential geometry". A submanifold M of a quaternionic Kahler manifold (M^^~, Q, g^^~) is said to be totally complex if there exists a section I^^~ of the bundle Q|_M such that (1)I^^~^2 = -id, (2)I^^~TM = TM (3)KTM ⊥ TM for any K ∈ Q|_M with <I^^~, K> = 0. Typical examples are half dimensional totally complex submanifolds of a quaternion projective space HP^n with parallel second fundamental form, which have been classified by Tsukada(head investigator of this research). The twistor space Z of (M^^~,Q,g^^~) is defined by Z = {I^^~ ∈ Q|I^^~^2 =-id}, which is an S^2-bundle over M^^~. The twistor space Z has a natural complex structure and it admits an Einstein-Kahler metric if M^^~ has positive scalar curvature.Main results of this research are the following :1.We show fundamental theorem on the existence and the uniqueness for half dimensional totally complex submanifolds of HP^n or the quaternion hyperbolic space HH^n. This result is an affirmative answer to the conjecture by Alekseevsky and Marchiafava.2.For a totally complex submanifold M of M^^~, we consider a new natural lift to the twistor space Z of M^^~ and construct a totally real and minimal submanifold of Z.3.We characterize half dimensional totally complex submanifolds of HP^n with parallel second fundamental form under some curvature condition such as Einstein-Kahler.4.We investigate fundamental properties of isotropic Kahler submanifolds of a complex quadric, whose theory is analogous to that of totally complex submanifolds.

在这项研究中，我们研究全复的四元数射影空间（更一般的四元数Kahler流形）的子流形。它是四元数微分几何和复微分几何相互作用的一个方面，并使所谓的“四元数复微分几何”。四元数Kahler流形（M^^~，Q，g^^~）的子流形M称为全复的，如果存在丛Q的截面I^^~|_M使得（1）I^^~^2 = -id，（2）I^^~TM = TM（3）KTM ≠ TM，对任意K ∈ Q|其中，K> = 0。典型的例子是具有平行第二基本形式的四元数射影空间HP^n的半维全复子流形，它们已经被Tsukada（本研究的首席研究员）分类。（M^^~，Q，g^^~）的扭量空间Z定义为Z = {I^^~ ∈ Q| I^^~^2 =-id}，它是M^^~上的S^2-丛。扭量空间Z具有自然的复结构，如果M^^~具有正的数量曲率，则Z具有Einstein-Kahler度量.本文的主要结果如下：1.给出了HP^n或四元数双曲空间HH^n的半维全复子流形的存在唯一性的基本定理.这一结果是对Alekseevsky和Marchiafava猜想的肯定回答。2.对于M^^~的全复子流形M，考虑M^^~的扭量空间Z的一个新的自然提升，构造了Z的一个全真实的极小子流形。3.在某些曲率条件下，如Einstein-4.研究了复二次曲面的迷向Kahler子流形的基本性质，其理论与全复子流形的理论类似。