Arc curvature in metric spaces

✍ Scribed by David C. Kay

• Publisher: Springer
• Year: 1980
• Tongue: English
• Weight: 556 KB
• Volume: 9
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00156477

✦ Synopsis

Concepts for curvature of arcs in metric geometry (specifically, Menger curvature KM, Haantjes-Finsler curvature Kn, and transverse curvature '~r introduced earlier by the author) are compared with respect to existence and numerical values. If a metric space satisfies a certain metric inequality shared in particular by Riemannian spaces, then the pointwise existence of ,cM on any arc implies that of '~r and the two are equal. In a Minkowskian plane X with strictly convex unit sphere whose boundary U has a C 2 polar representation p = p(O), and with ~M and '~r the Menger and transverse curvatures relative to the underlying Euclidean metric, the following formulas are proved: At any point p on an arc at which ~M and '~r exist,

where Tp is the tangent at p, T~ that line to which Tp is metrically perpendicular, and a~ and or2 are certain real-valued functions defined on lines of X. The result of this is that if ,~* is the classical curvature of U~ -U + p at U~ c~ Tp, K 2 = '¢*°~t2(T~)

K~ e2(T,, T~)'

from which it follows that the values of ,¢M and KT are not equal for metric spaces in general even when both exist.