Woit · §6.2 Spin groups in three and four dimensions
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英文 OCR 草稿
The source identifies SO(3) with S^3 modulo antipodal points and describes RP^3 as lines through the origin in R^4; role is discussion.
机器草稿中的相关讨论摘录
Both u and − act in the same way on so we have two elements in corresponding to the same element in SO(3). One can show that is a surjective map (any element of SO(3) is of something), so it is what is called a “covering” map, specifically a two-fold cover. It makes a double cover of , and we give this group the name . This also allows us to characterize more simply as a geometrical space. It is with opposite points on the three-sphere identified. This space is known as , real projective 3-space, which can also be thought of as the space of lines through the origin in (each such line intersects in two opposite points).
中文机译草稿
本处按 PDF 核对术语及讨论角色；OCR 和机译仍为未审校机器稿。
机器草稿中的相关讨论摘录
与 − 对 的作用方式相同，因此在 中有两个元素对应于 SO(3) 中的同一个元素。可以证明 是满射（SO(3) 的任意元素都是某个元素的 ），因此它就是所谓的 a“覆盖”映射，具体地是一个二重覆盖。它使 成为 的双重覆盖，我们将这个群命名为 。这也使我们能够更简单地将 刻画为一个几何空间。它就是 ，其中三球面上的对径点被等同。这个空间被称为 ，即实射影 3-空间，它也可以被视为 中过原点的直线构成的空间（每条这样的直线与 交于两个对径点）。
来源与版本
正文：中文机译草稿
来源版本：2025-10-20
原文位置：§6.2；PDF 页 85；印刷页 68
处理记录：MinerU OCR；machine-translation/deepseek-v4.1-flash
核对状态：中文机译草稿 · 待校对
Woit · §7.4 Complex projective space
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英文 OCR 草稿
The author generalizes the line model to RP^{n-1} and recalls the identification SO(3)=RP^3.
机器草稿中的相关讨论摘录
In higher dimensions, the space of lines through the origin in is called and can be thought of as the unit sphere in , with opposite points identified (recall from section 6.2.3 that can be identified with .