In Riemannian geometry, the geodesic curvature k g {\displaystyle k_{g}} of a curve γ {\displaystyle \gamma } measures how far the curve is from being a geodesic. For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of the curve projected onto the surface's tangent plane. More generally, in a given manifold M ¯ {\displaystyle {\bar {M}}} , the geodesic curvature is just the usual curvature of γ {\displaystyle \gamma } (see below). However, when the curve γ {\displaystyle \gamma } is restricted to lie on a submanifold M {\displaystyle M} of M ¯ {\displaystyle {\bar {M}}} (e.g. for curves on surfaces), geodesic curvature refers to the curvature of γ {\displaystyle \gamma } in M {\displaystyle M} and it is different in general from the curvature of γ {\displaystyle \gamma } in the ambient manifold M ¯ {\displaystyle {\bar {M}}} . The (ambient) curvature k {\displaystyle k} of γ {\displaystyle \gamma } depends on two factors: the curvature of the submanifold M {\displaystyle M} in the direction of γ {\displaystyle \gamma } (the normal curvature k n {\displaystyle k_{n}} ), which depends only on the direction of the curve, and the curvature of γ {\displaystyle \gamma } seen in M {\displaystyle M} (the geodesic curvature k g {\displaystyle k_{g}} ), which is a second order quantity. The relation between these is k = k g 2 + k n 2 {\displaystyle k={\sqrt {k_{g}^{2}+k_{n}^{2}}}} . In particular geodesics on M {\displaystyle M} have zero geodesic curvature (they are "straight"), so that k = k n {\displaystyle k=k_{n}} , which explains why they appear to be curved in ambient space whenever the submanifold is.
Definition Consider a curve γ {\displaystyle \gamma } in a manifold M ¯ {\displaystyle {\bar {M}}} , parametrized by arclength, with unit tangent vector T = d γ / d s {\displaystyle T=d\gamma /ds} . Its curvature is the norm of the covariant derivative of T {\displaystyle T} : k = ‖ D T / d s ‖ {\displaystyle k=\|DT/ds\|} . If γ {\displaystyle \gamma } lies on M {\displaystyle M} , the geodesic curvature is the norm of the projection of the covariant derivative D T / d s {\displaystyle DT/ds} on the tangent space to the submanifold. Conversely the normal curvature is the norm of the projection of D T / d s {\displaystyle DT/ds} on the normal bundle to the submanifold at the point considered. If the ambient manifold is the euclidean space R n {\displaystyle \mathbb {R} ^{n}} , then the covariant derivative D T / d s {\displaystyle DT/ds} is just the usual derivative d T / d s {\displaystyle dT/ds} . If γ {\displaystyle \gamma } is unit-speed, i.e. ‖ γ ′ ( s ) ‖ = 1 {\displaystyle \|\gamma '(s)\|=1} , and N {\displaystyle N} designates the unit normal field of M {\displaystyle M} along γ {\displaystyle \gamma } , the geodesic curvature is given by
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