Difference between revisions of "Internal metric"
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| − | A [[Metric|metric]] $\rho$ which is defined for any two points $x$, $y$ of a [[Metric space|metric space]] that can be connected by a rectifiable curve $\gamma(x,y)$ and for which | + | A [[Metric|metric]] $\rho$ which is defined for any two points $x$, $y$ of a [[Metric space|metric space]] that can be connected by a [[rectifiable curve]] $\gamma(x,y)$ and for which |
$$\rho(x,y)=\inf_\gamma s_\rho(\gamma(x,y)),$$ | $$\rho(x,y)=\inf_\gamma s_\rho(\gamma(x,y)),$$ | ||
| − | 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 | + | 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 |
$$\rho^*(x,y)=\inf_\gamma s_\rho(\gamma(x,y))$$ | $$\rho^*(x,y)=\inf_\gamma s_\rho(\gamma(x,y))$$ | ||
Latest revision as of 12:31, 9 November 2014
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) |
Comments
The internal metric is better known as the interior metric, and is nearly the same as the Western use of the phrase "convex metric"; for the (stronger) notion of "convex metric" as used in the Soviet Union, see Convex metric.
For the notion of "internal metric" as used in the general theory of metric spaces, see Metric.
References
| [a1] | W. Rinow, "Die innere Geometrie der metrischen Räume" , Springer (1961) |
Internal metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Internal_metric&oldid=34399