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.
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For a unit-speed curve on an oriented surface with unit normal , let and let denote tangential covariant differentiation. Its signed geodesic curvature isIt vanishes exactly when the curve is a geodesic.