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.
If ambient isometries preserving a connected curve act transitively on it, then the magnitude of its geodesic curvature is constant. Continuity then makes the signed geodesic curvature either identically zero or of one fixed sign.
If a closed oriented surface is cut along a curve into two isometric surfaces with boundary, their Euler characteristics and total Gaussian curvatures agree. Applying Gauss-Bonnet to both halves, whose induced boundary orientations are opposite, gives
A non-equatorial latitude on the round sphere is preserved transitively by rotations about its axis but has nonzero geodesic curvature. Its two complementary spherical caps are not isometric.
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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.