Geodesic Metric Space

A geodesic metric space is a metric space in which any two points are joined by a path isometric to the real interval (Bridson and Haefliger 1999, p. 4). Such a path realizes the distance between its endpoints. This condition requires a globally distance-minimizing path, beyond local minimization along a curve.

For example, Euclidean space and a connected graph with each edge realized as an interval of length 1, equipped with the path metric, are geodesic metric spaces. The comparison condition for a CAT(0) space further controls distances between points on these paths.

See also

Alexandrov Space, CAT(0) Space, Distance, Geodesic, Length Metric, Length Space, Metric Space, RCD Space

Explore with Wolfram|Alpha

More things to try:

References

Bridson, M. R. and Haefliger, A. Metric Spaces of Non-Positive Curvature. Berlin, Germany: Springer-Verlag, 1999. https://doi.org/10.1007/978-3-662-12494-9.

Cite this as:

Weisstein, Eric W. "Geodesic Metric Space." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GeodesicMetricSpace.html

Subject classifications