Arc curvature in metric spaces

Concepts for curvature of arcs in metric geometry (specifically, Menger curvature κ _M , Haantjes-Finsler curvature κ _H , and transverse curvature κ _T 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 κ _M on any arc implies that of κ _T and the two are equal. In a Minkowskian plane X with strictly convex unit sphere whose boundary U has a C ² polar representation ρ=ρ(θ), and with (bar kappa _M) and (bar kappa _M) 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 (bar kappa _M) and (bar kappa _M) exist, $$kappa _M = bar kappa _M sqrt {rho ^{2 + } 2rho ^{'2 - } rho rho }$$ and $$kappa _T = bar kappa _T frac{{sigma _1^{3/2} (T_p )}}{{sigma _2 (T_p ,T_p^ bot )}},$$ where T _pis the tangent at p, T ^⊥ _pthat line to which T _pis metrically perpendicular, and σ₁ and σ₂ are certain real-valued functions defined on lines of X. The result of this is that if κ* is the classical curvature of U _p≡U+p at U _p∩ T _p, $$frac{{kappa _M^2 }}{{kappa _T^2 }} = frac{{kappa ^ * sigma _1^{3/2} (T_p^ bot )}}{{sigma _2 left( {T_p ,T_p^ bot } right)}},$$ from which it follows that the values of κ _M and κ _T are not equal for metric spaces in general even when both exist.