nLab length

Idea

Length is the volume of curves.

Definition

Let $(X,d)$ be a metric space, and let $c:[0,1]\to X$ be a continuous curve. The length of $c$ is the number (possibly infinite) given by

$$|c| \;\coloneqq\; \sup_{n, t_0,\dots,t_n} \sum_{i=1}^n d\big( c(t_{i-1}), c(t_i)\big),$$

where the supremum is taken over all finite partitions of the interval $0=t_0 \le t_1\le \dots\le t_n =1$.

For Riemannian manifolds

If $X$ is a Riemannian manifold (with the induced metric) and $c$ is a smooth curve, the length of $c$, sometimes called the arc length, is equivalently given by the formula

$$|c| \;=\; \int_0^1 |c'(t)|\,d t,$$

where $c'(t)$ is the tangent vector, and $|c'(t)|$ its length, given in terms of the Riemannian metric.

Examples

• circumference of a circle

Length and geodesic spaces

In many situations, both in geometry and in ordinary life, the distance between two points is the length of the shortest possible path between them. Here is how one can make this mathematically precise:

• A metric space $(X,d)$ is called a length space if and only if for all $x,y\in X$,
  $$d(x,y) \;=\; \inf_{c:[0,1]\to X} |c| ,$$

  where the infimum is taken over all continuous curves $c:[0,1]\to X$. Note that the equality above may fail in general, take for example the metric on the circle induced by its embedding into the plane. (The inequality $\le$ always holds.)

• $(X,d)$ is moreover called a geodesic space if the infimum above is always attained (and a curve at which it is attained is called a geodesic).

Related concepts

• area

• volume

• curve

• length scale

• wave length

• Planck length

• Compton wavelength

• string length