linguistic:translation
Riemannian metrics and Riemannian manifolds

Let M {\displaystyle M} be a smooth manifold. For each point p ∈ M {\displaystyle p\in M}, there is an associated vector space T p M {\displaystyle T_{p}M} called the tangent space of M {\displaystyle M} at p {\displaystyle p}. Vectors in T p M {\displaystyle T_{p}M} are thought of as the vectors tangent to M {\displaystyle M} at p {\displaystyle p}.

However, T p M {\displaystyle T_{p}M} does not come equipped with an inner product, a measuring stick that gives tangent vectors a concept of length and angle. This is an important deficiency because calculus teaches that to calculate the length of a curve, the length of vectors tangent to the curve must be defined. A Riemannian metric puts a measuring stick on every tangent space.

A Riemannian metric g {\displaystyle g} on M {\displaystyle M} assigns to each p {\displaystyle p} a positive-definite inner product g p: T p M × T p M → R {\displaystyle g_{p}:T_{p}M\times T_{p}M\to \mathbb {R} } in a smooth way (see the section on regularity below). This induces a norm ‖ ⋅ ‖ p: T p M → R {\displaystyle \|\cdot \|_{p}:T_{p}M\to \mathbb {R} } defined by ‖ v ‖ p = g p (v, v) {\displaystyle \|v\|_{p}={\sqrt {g_{p}(v,v)}}}. A smooth manifold M {\displaystyle M} endowed with a Riemannian metric g {\displaystyle g} is a Riemannian manifold, denoted (M, g) {\displaystyle (M,g)}. A Riemannian metric is a special case of a metric tensor.

A Riemannian metric is not to be confused with the distance function of a metric space, which is also called a metric.