按作者与版本查阅
以下按已登记位置排列。出现顺序、原文引用与已登记依赖分别展示；最早位置不代表全书首次定义。
Szekeres · source-snapshot-bc3404156f6e
已登记出处中的最早位置：§7.2 · zh-CN。
来源短摘录与定位
§7.2 · zh-CN · translation · unreviewed
例 7.5
设 为实内积空间，如 第 5.1 节所述。由下式定义的映射
显然是双线性的，并且是一个2阶协变张量，称为内积的度量张量(metric tensor)。内积的分量是度量张量关于基的分量
而两个向量的内积是
度量张量是对称的，
§7.2 · en · original_transcription · unreviewed
Example 7.5
Let be a real inner product space, as in Section 5.1. The map defined by
is obviously bilinear, and is a covariant tensor of degree 2 called the metric tensor of the inner product. The components of the inner product are the components of the metric tensor with respect to the basis
while the inner product of two vectors is
The metric tensor is symmetric,
§18.4 · zh-CN · translation · unreviewed
流形 上类型为 (0, 2) 的张量场 称为非奇异的，如果 在每一点 处都是非奇异张量。一个 伪黎曼流形(pseudo-Riemannian manifold) 由一个可微流形 连同类型为 (0, 2) 的对称非奇异张量场 组成，该张量场称为 度量张量(metric tensor)。这等价于在每一点 处的切空间 上定义一个内积 （见第 5 章和第 7 章）。我们将假设 是可微张量场，从而对于每一对光滑向量场 、，其内积都是可微函数，
§18.4 · en · original_transcription · unreviewed
A tensor field of type (0, 2) on a manifold is said to be non-singular if is a non-singular tensor at every point . A pseudo-Riemannian manifold consists of a differentiable manifold together with a symmetric non-singular tensor field of type (0, 2), called a metric tensor. This is equivalent to defining an inner produc on the tangent space at every point (see Chapters 5 and 7). We will assume is a differentiable tensor field, so that for every pair of smooth vector fields , the inner product is a differentiable function,