Intrinsic metric

In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second along a path whose arclength is equal to (or very close to) that distance. The distance between two points of a metric space relative to the intrinsic metric is defined as the infimum of the lengths of all paths from the first point to the second. A metric space is a length metric space if the intrinsic metric agrees with the original metric of the space.

If the space has the stronger property that there always exists a path that achieves the infimum of length (a geodesic) then it is called a geodesic metric space or geodesic space. For instance, the Euclidean plane is a geodesic space, with line segments as its geodesics. The Euclidean plane with the origin removed is not geodesic, but is still a length metric space.

Let ${\displaystyle (M,d)}$ be a metric space, i.e., ${\displaystyle M}$ is a collection of points (such as all of the points in the plane, or all points on the circle) and ${\displaystyle d(x,y)}$ is a function that provides us with the distance between points ${\displaystyle x,y\in M}$. We define a new metric ${\displaystyle d_{\text{I}}}$ on ${\displaystyle M}$, known as the induced intrinsic metric, as follows: ${\displaystyle d_{\text{I}}(x,y)}$ is the infimum of the lengths of all paths from ${\displaystyle x}$ to ${\displaystyle y}$.

Here, a path from ${\displaystyle x}$ to ${\displaystyle y}$ is a continuous map

with ${\displaystyle \gamma (0)=x}$ and ${\displaystyle \gamma (1)=y}$. The length of such a path is defined as follows: to each finite partition

of the interval ${\displaystyle [0,1]}$, consider the sum

We then define the length of ${\displaystyle \gamma }$ to be

where ${\displaystyle {\mathfrak {P}}}$ is the set of finite partitions of ${\displaystyle [0,1]}$. If the supremum is finite, we call ${\displaystyle \gamma }$ a rectifiable curve. Note that ${\displaystyle d_{\text{I}}(x,y)=\infty }$ if there is no path from ${\displaystyle x}$ to ${\displaystyle y}$ since the infimum of the empty set within the closed interval [0,+∞] is +∞.