Totally complex submanifolds of a quaternion projective space

在本研究中，我们研究了四元数投影空间的完全复子流形（更一般地说是四元数Kahler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，形成了所谓的“四元数复微分几何”。四元数Kahler流形（M^^~， Q, g^^~）的子流形M被称为完全复流形，如果存在一个集Q|_M的I^^~区间，使得(1)I^^~ = -id, (2)I^^~TM = TM(3)对于任意K∈Q|_M <I^^~， K> = 0, KTM⊥TM。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，它已被Tsukada（本研究的首席研究员）分类。（M^^~，Q,g^^~）的扭转空间Z定义为Z = {I^^~∈Q|I^^~^2 =-id}，它是M^^~上的一个S^2束。扭转空间Z具有自然的复杂结构，如果M^^~具有正的标量曲率，则它允许爱因斯坦-卡勒度规。本研究的主要成果如下：给出了四元双曲空间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。对于M^^~的完全复子流形M，考虑M^^~的扭曲空间Z的一个新的自然提升，构造了Z.3的完全实数极小子流形。在某些曲率条件下，我们刻画了具有平行第二基本形式的半维全复HP^n子流形。研究了复二次型各向同性Kahler子流形的基本性质，其理论与完全复子流形的理论类似。

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.