 Geodesic curvature

For a unit-speed curve $\alpha$ on an oriented surface with unit normal $N$, let $T=\dot{\alpha }$ and let $D_s$ denote tangential covariant differentiation. Its signed geodesic curvature isIt vanishes exactly when the curve is a geodesic.

Geodesic curvature is a concept from differential geometry that pertains to the curvature of curves on surfaces. More specifically, it measures how much a given curve deviates from being a geodesic on a surface. To understand geodesic curvature, it's helpful to first define some basic terms: 1. **Geodesic**: A geodesic is the shortest path between two points on a surface. On a flat surface, geodesics are straight lines.