# Internal metric

A metric $\rho$ which is defined for any two points $x$, $y$ of a metric space that can be connected by a rectifiable curve $\gamma(x,y)$ and for which

$$\rho(x,y)=\inf_\gamma s_\rho(\gamma(x,y)),$$

where $s_\rho$ is the length of the curve in the metric $\rho$. A Riemannian metric induces an internal metric. If, in a space with a metric $\rho$, any two points may be connected by a rectifiable curve, the equality

$$\rho^*(x,y)=\inf_\gamma s_\rho(\gamma(x,y))$$

defines an internal metric, and serves as the definition of the internal metric $\rho^*$ induced on a manifold immersed in this metric space.

#### References

 [1] A.D. Aleksandrov, "Die innere Geometrie der konvexen Flächen" , Akademie Verlag (1955) (Translated from Russian)