Difference between revisions of "Internal metric"
From Encyclopedia of Mathematics
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| − | A [[Metric|metric]] | + | {{TEX|done}} |
| + | 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 | ||
| − | + | $$\rho(x,y)=\inf_\gamma s_\rho(\gamma(x,y)),$$ | |
| − | where | + | 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 | + | defines an internal metric, and serves as the definition of the internal metric $\rho^*$ induced on a manifold immersed in this metric space. |
====References==== | ====References==== | ||
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====Comments==== | ====Comments==== | ||
| − | The internal metric is better known as the interior metric, and is nearly the same as the Western use of the phrase | + | 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|Convex metric]]. |
| − | For the notion of | + | For the notion of "internal metric" as used in the general theory of metric spaces, see [[Metric|Metric]]. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rinow, "Die innere Geometrie der metrischen Räume" , Springer (1961)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rinow, "Die innere Geometrie der metrischen Räume" , Springer (1961)</TD></TR></table> | ||
Revision as of 11:06, 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) |
How to Cite This Entry:
Internal metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Internal_metric&oldid=16462
Internal metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Internal_metric&oldid=16462
This article was adapted from an original article by Yu.D. Burago (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article