Convexity of reflective submanifolds in symmetric $R$-spaces
Convexity of reflective submanifolds in symmetric $R$-spaces
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对称 $R$-空间中反射子流形的凸性
DOI：
10.2748/tmj/1356038981
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发表时间：
2012
影响因子：
0.5
通讯作者：
Makiko Tanaka
中科院分区：
文献类型：
--
作者：
Peter Quast;Makiko Tanaka
我们证明了对称R-空间的每个反射子流形都是（测地线）凸的。导论.本文的主要结果如下。定理1.对称R-空间的反射子流形是（测地线）凸的。我们组织这篇文章如下。在第1节中，我们定义了定理1中使用的所有概念。对称R-空间中的反射子流形在第2节中描述。定理1的证明可以在第3节中找到。在第四节中，我们解释了定理1中的“对称R-空间”假设不能推广到所有紧对称空间的原因。对称R-空间，由Takeuchi和Nagano在1960年代引入，形成了一类紧致对称空间，具有非常特殊的几何性质，并出现在各种环境中。例如，对称R-空间作为某些最短测地线的空间出现，即作为那些由连接基点和极点的最短测地线弧的中点形成的中心粒（参见[CN 88]）（参见例如[MQ 12]）。对称空间中的反射子流形包括极和中心粒（参见[CN 88，Na 88，Qu 11]）。Bott [Bo 59]在他关于经典李群的同伦群的著名周期性结果的第一个证明中使用了涉及这种中心粒的迭代构造（参见[Mi69，§ 23，24]和[Mi 88，§ 7]）。对于[MQ 11，Sect. 1.2]中描述的构造，重要的是在中心粒中测量的基点和某些R-空间的中心粒中的极点之间的距离与在周围R-空间中测量的距离相同。这直接由定理1得出。定理1还在周围空间是对称R-空间的情况下给出了[NS 91，注3.2b]的概念性证明。1.你的助手。我们首先定义定理1中使用的术语。反射子流形黎曼流形P的一个反射子流形M是P的一个对合等距τ的不动点集的连通分支，即τ 2等于2000数学主题分类。小学53 C35;中学53 C40。
We show that every reflective submanifold of a symmetric R-space is (geodesically) convex. Introduction. The main result in this article is the following. THEOREM 1. Reflective submanifolds of symmetric R-spaces are (geodesically) convex. We organized this article as follows. In Section 1, we define all notions used in Theorem 1. Reflective submanifolds in symmetric R-spaces are described in Section 2. The proof of Theorem 1 can be found in Section 3. In Section 4, we explain why the assumption “symmetric R-space” in Theorem 1 can not be generalized to all compact symmetric spaces. Symmetric R-spaces, introduced by Takeuchi and Nagano in the 1960s, form a class of compact symmetric spaces that have very peculiar geometric properties and appear in various contexts. For example, symmetric R-spaces arise as certain spaces of shortest geodesics, namely as those centrioles (see [CN88]) that are formed by midpoints of shortest geodesics arcs joining a base point to a pole (see e.g. [MQ12]). Reflective submanifolds in symmetric spaces include among others polars and centrioles (see e.g. [CN88, Na88, Qu11]). An iterative construction involving such centrioles has been used by Bott [Bo59] in the first proof of his famous periodicity result for the homotopy groups of classical Lie groups (see also [Mi69, § 23, 24] and [Mi88, § 7]). For the construction described in [MQ11, Sect. 1.2], it is important that the distance between a base point and a pole in a centriole of certain R-spaces measured in the centriole is the same as the distance measured in the ambient R-space. This follows directly from Theorem 1. Theorem 1 also provides a conceptional proof of [NS91, Remark 3.2b] in the case where the ambient space is a symmetric R-space. 1. Preliminaries. We first define the terminology used in Theorem 1. Reflective submanifolds. A reflective submanifold M of a Riemannian manifold P is a connected component of the fixed point set of an involutive isometry τ of P, that is τ 2 equals 2000 Mathematics Subject Classification. Primary 53C35; Secondary 53C40.
影响因子：
1.1
作者：
D. Leung
通讯作者：
D. Leung