# Geodesic curvature

In Riemannian geometry, the geodesic curvature ${\displaystyle k_{g}}$ of a curve ${\displaystyle \gamma }$ measures how far the curve is from being a geodesic. In a given manifold ${\displaystyle {\bar {M}}}$, the geodesic curvature is just the usual curvature of ${\displaystyle \gamma }$ (see below), but when ${\displaystyle \gamma }$ is restricted to lie on a submanifold ${\displaystyle M}$ of ${\displaystyle {\bar {M}}}$ (e.g. for curves on surfaces), geodesic curvature refers to the curvature of ${\displaystyle \gamma }$ in ${\displaystyle M}$ and it is different in general from the curvature of ${\displaystyle \gamma }$ in the ambient manifold ${\displaystyle {\bar {M}}}$. The (ambient) curvature ${\displaystyle k}$ of ${\displaystyle \gamma }$ depends on two factors: the curvature of the submanifold ${\displaystyle M}$ in the direction of ${\displaystyle \gamma }$ (the normal curvature ${\displaystyle k_{n}}$), which depends only from the direction of the curve, and the curvature of ${\displaystyle \gamma }$ seen in ${\displaystyle M}$ (the geodesic curvature ${\displaystyle k_{g}}$), which is a second order quantity. The relation between these is ${\displaystyle k={\sqrt {k_{g}^{2}+k_{n}^{2}}}}$. In particular geodesics on ${\displaystyle M}$ have zero geodesic curvature (they are "straight"), so that ${\displaystyle k=k_{n}}$, which explains why they appear to be curved in ambient space whenever the submanifold is.

## Definition

Consider a curve ${\displaystyle \gamma }$ in a manifold ${\displaystyle {\bar {M}}}$, parametrized by arclength, with unit tangent vector ${\displaystyle T=d\gamma /ds}$. Its curvature is the norm of the covariant derivative of ${\displaystyle T}$: ${\displaystyle k=\|DT/ds\|}$. If ${\displaystyle \gamma }$ lies on ${\displaystyle M}$, the geodesic curvature is the norm of the projection of the covariant derivative ${\displaystyle DT/ds}$ on the tangent space to the submanifold. Conversely the normal curvature is the norm of the projection of ${\displaystyle DT/ds}$ on the normal bundle to the submanifold at the point considered.

If the ambient manifold is the euclidean space ${\displaystyle \mathbb {R} ^{n}}$, then the covariant derivative ${\displaystyle DT/ds}$ is just the usual derivative ${\displaystyle dT/ds}$.

## Example

Let ${\displaystyle M}$ be the unit sphere ${\displaystyle S^{2}}$ in three-dimensional Euclidean space. The normal curvature of ${\displaystyle S^{2}}$ is identically 1, independently of the direction considered. Great circles have curvature ${\displaystyle k=1}$, so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius ${\displaystyle r}$ will have curvature ${\displaystyle 1/r}$ and geodesic curvature ${\displaystyle k_{g}={\sqrt {1-r^{2}}}/r}$.

## References

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