曲率

本书暂定概念：尚未进入正式概念库。下面是本书给出的定义原文。

定义 · 7.3 曲率与挠率

在进一步讨论之前，我们先考察这些张量的几何意义。先看黎曼张量。一个关键的观察是：若把 $p$ 处的向量 $V$ 沿两条不同的曲线 $C$ 和 $C^{\prime }$ 作平行移动到 $q$，则一般说来在 $q$ 处得到的向量并不相同（图 7.2）。然而，若在欧几里得空间中作平行移动——那里的平行移动按我们通常的意义定义——则所得向量与所沿的路径无关。我们期望，正是平行移动的这种不可积性刻画了内禀的曲率(curvature)概念，而这一概念并不依赖于

英文原文

Before we proceed further, we examine the geometrical meaning of these tensors. We consider the Riemann tensor first. A crucial observation is that if we parallel transport a vector $V$ at $p$ to $q$ along two different curves $C$ and $C^{\prime }$ , the resulting vectors at $q$ are different in general (figure 7.2). If, however, we parallel transport a vector in a Euclidean space, where the parallel transport is defined in our usual sense, the resulting vector does not depend on the path along which it has been parallel transported. We expect that this non-integrability of parallel transport characterizes the intrinsic notion of curvature, which does not depend

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