I feel it necessary to point out that the term "curvature" applied to 1-dimensional curves is a different notion to the term "curvature" applied to spaces of 2 or more dimensions. Under the assumption that the 1-dimensional curve is embedded in a higher-dimensional space, "curvature" is defined as the absolute derivative of the unit tangent vector with respect to the arc-length parameter along the curve:

where

is the unit tangent vector (

is the arc-length parameter along the curve)

All values,

,

, and

are defined in terms of the embedding space. By contrast, for spaces of 2 or more dimensions, the term "curvature", when used without qualification, is defined as the Riemann curvature tensor field:

(It should be noted that different texts define the Riemann curvature tensor differently, resulting in a different sign, particularly in the Einstein equation)