# Geodesic curvature

In Riemannian geometry, the geodesic curvature $k_g$ of a curve lying on a submanifold of the ambient space measures how far the curve is from being a geodesic. (For instance it applies to curves on surfaces.) The notion of geodesic curvature allows to distinguish the part of the curvature in ambient space that is due to the submanifold (the normal curvature $k_n$) and the one that comes from the curve itself. The curvature $k$ of the curve is related to these two by $k = \sqrt{k_g^2+k_n^2}$. In particular geodesics have zero geodesic curvature (they are "straight"), and that is their definition, so that $k=k_n$, which explains why they appear to be curved in ambient space whenever the submanifold is.

## Definition

Consider a curve lying on a submanifold $M$ in ambient manifold $\bar{M}$, parametrized by arclength $s$, with unit tangent vector $T$. The geodesic curvature is the norm of the projection of the derivative $dT/ds$ on the tangent plane to the submanifold. Conversely the normal curvature is the norm of the projection of $dT/ds$ on the normal bundle to the submanifold at the point considered.

## Example

Let $M$ be the unit sphere $S^2$ in three dimensional Euclidean space. The normal curvature of $S^2$ is identically 1. Great circles have curvature $k=1$, which implies zero geodesic curvature, thus they are geodesics. Smaller circles of radius $r$ will have curvature $1/r$ and geodesic curvature $k_g = \sqrt{1-r^2}/r$.

## Some results involving geodesic curvature

• The geodesic curvature is no other than the usual curvature of the curve when computed intrinsically in the submanifold $M$. It does not depend on the way the submanifold $M$ sits in $\bar{M}$.
• On the other hand the normal curvature depends strongly on how the submanifold lies in the ambient space, but marginally on the curve: $k_n$ only depends on the point on the submanifold and the direction $T$, but not on $dT/ds$.
• In general Riemannian geometry the derivative will be computed using the Levi-Civita connection $\bar{\nabla}$ of the ambient manifold. It splits into a tangent part and a normal part to the submanifold: $\bar{\nabla}_T T = \nabla_T T + (\bar{\nabla}_T T)^\perp$, and the tangent part is then the usual derivative $\nabla_T T$ in $M$ (see Gauss equation in the Gauss-Codazzi equations).