Geodesic Curvature

For a unit speed curve on a surface, the length of the surface-tangential component of acceleration is the geodesic curvature . Curves with are called Geodesics. For a curve parameterized as , the geodesic curvature is given by

where , , and are coefficients of the first Fundamental Form and are Christoffel Symbols of the Second Kind.

Gray, A. Geodesic Curvature.'' §20.5 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 402-407, 1993.