Convex metric
An internal metric on a two-dimensional manifold which meets some convexity condition. More exactly, let
and
be two shortest lines issuing from some point
; let
and
be points on these lines; let
be the distances from
to
and
, respectively; let
be the distance between
and
; and let
be the angle opposite to the side
in the plane triangle with sides
. The convexity condition of the metric (at the point
) is that
is a non-increasing function (i.e.
if
,
) on any pair of intervals
,
such that the points
and
, which correspond to two arbitrary values in these intervals, can be connected by a shortest line. An internal metric is a convex metric if and only if it is a metric of non-negative curvature. The metric of a convex surface is a convex metric. Conversely, any two-dimensional manifold with a convex metric can be realized as a convex surface (Aleksandrov's theorem).
References
[1] | A.D. Aleksandrov, "Die innere Geometrie der konvexen Flächen" , Akademie Verlag (1955) (Translated from Russian) |
Convex metric. M.I. Voitsekhovskii (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convex_metric&oldid=12734