Geodesic curvature is a concept in differential geometry that quantifies how a curve on a surface deviates from being a geodesic, i.e., the shortest path between nearby points on that surface. While the curvature of a space curve in Euclidean space measures the rate of change of its tangent vector in all directions, the geodesic curvature isolates the component of this change that lies within the tangent plane of the surface on which the curve resides.
Formally, let $M$ be an oriented regular surface in $\mathbb{R}^3$ with unit normal field $\mathbf{N}$. For a smooth curve $\gamma(s)$ parametrized by arc length $s$ on $M$, the curvature vector $\kappa(s)\mathbf{n}(s) = \frac{d^2\gamma}{ds^2}$ can be decomposed into a component normal to $M$ and a component tangent to $M$: $$ \frac{d^2\gamma}{ds^2}=k_n \mathbf{N} + k_g \mathbf{t}_g, $$ where $k_n$ is the normal curvature, $k_g$ is the geodesic curvature, and $\mathbf{t}_g$ is a unit vector lying in the tangent plane of $M$ orthogonal to the curve’s tangent. Equivalently, $$ k_g = \langle \frac{d^2\gamma}{ds^2}, \mathbf{t}_g\rangle, $$ where $\langle\cdot,\cdot\rangle$ denotes the Euclidean inner product.
Geodesic curvature has several important properties and applications:
- Geodesic characterization: A curve on a surface is a geodesic if and only if its geodesic curvature vanishes everywhere ($k_g\equiv0$). Thus, geodesic curvature provides a diagnostic tool for identifying geodesics.
- Relation to total curvature: For a closed curve $C$ on an oriented surface, the Gauss–Bonnet theorem links the integral of geodesic curvature along $C$ with the total Gaussian curvature of the region it bounds: $$ \int_C k_g , ds + \int_{\Omega} K , dA = 2\pi \chi(\Omega), $$ where $K$ is the Gaussian curvature of the surface, $dA$ the area element, $\chi(\Omega)$ the Euler characteristic of the region $\Omega$, and $ds$ the line element along $C$.
- Physical contexts: In mechanics and optics, geodesic curvature appears in the analysis of constrained motion on surfaces and in the study of light rays on curved interfaces, where the deviation from geodesic paths influences forces and trajectories.
- Computational formulas: When a surface is expressed in local coordinates $(u^1,u^2)$ with metric tensor $g_{ij}$ and Christoffel symbols $\Gamma^k_{ij}$, the geodesic curvature of a curve $\gamma(t)=(u^1(t),u^2(t))$ can be computed via $$ k_g = \frac{1}{|\dot\gamma|}\Bigl| abla_{\dot\gamma}\dot\gamma - ( abla_{\dot\gamma}\dot\gamma\cdot\mathbf{N})\mathbf{N}\Bigr|, $$ where $ abla$ denotes the Levi‑Civita connection of the surface.
The concept extends to higher‑dimensional manifolds, where the geodesic curvature of a curve is defined relative to the ambient Levi‑Civita connection and the tangent subspace of the submanifold containing the curve. In such contexts, it continues to measure the failure of the curve to be a geodesic of the submanifold.
