# Geodesic curvature

**curvature vectorgeodesic curvespath curvature**

In Riemannian geometry, the geodesic curvature k_g of a curve \gamma measures how far the curve is from being a geodesic.wikipedia

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### Geodesic

**geodesicsgeodesic flowgeodesic equation**

In Riemannian geometry, the geodesic curvature k_g of a curve \gamma measures how far the curve is from being a geodesic.

In a Riemannian manifold or submanifold geodesics are characterised by the property of having vanishing geodesic curvature.

### Curvature

**curvednegative curvatureextrinsic curvature**

For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of the curve projected onto the surface's tangent plane.

These are the normal curvature, geodesic curvature and geodesic torsion.

### Gauss–Bonnet theorem

**Gauss-Bonnet theoremGauss–Bonnet formulaChern–Gauss–Bonnet formula**

Suppose M is a compact two-dimensional Riemannian manifold with boundary \partial M. Let K be the Gaussian curvature of M, and let k_g be the geodesic curvature of \partial M.

### Darboux frame

**normal curvaturegeodesic torsionDarboux basis**

The (ambient) curvature k of \gamma depends on two factors: the curvature of the submanifold M in the direction of \gamma (the normal curvature k_n), which depends only on the direction of the curve, and the curvature of \gamma seen in M (the geodesic curvature k_g), which is a second order quantity.

### Riemannian geometry

**Riemannianlocal to global theoremsRiemann geometry**

In Riemannian geometry, the geodesic curvature k_g of a curve \gamma measures how far the curve is from being a geodesic.

### Arc length

**rectifiable curvearclengthlength**

Consider a curve \gamma in a manifold \bar{M}, parametrized by arclength, with unit tangent vector.

### Covariant derivative

**covariant differentiationtensor derivativecovariant differential**

Its curvature is the norm of the covariant derivative of T:.

### Levi-Civita connection

**Christoffel symbolconnectionsLevi-Civita**

### Second fundamental form

**extrinsic curvaturesecondshape tensor**

### Gauss–Codazzi equations

**Gauss-Codazzi equationsGauss–Codazzi equationGauss–Codazzi equations (relativity)**

### Ferdinand Minding

**MindingMinding, Ferdinand**

Minding considered questions of bending of surfaces and proved the invariance of geodesic curvature.

### Hyperbolic geometry

**hyperbolic planehyperbolichyperbolic surface**

If the Gaussian curvature of the plane is −1 then the geodesic curvature of a circle of radius r is:

### Geodesics on an ellipsoid

**geodesicsEarth geodesicsgeodesic**

geodesic curvature—i.e., the analogue of straight lines on a

### Rhumb line

**loxodromerhumbconstant-rhumb trajectories**

In other words, a great circle is locally "straight" with zero geodesic curvature, whereas a rhumb line has non-zero geodesic curvature.

### Tensor network theory

**tensor networktensor network model**

This allowed the trajectories of objects to be modeled as geodesic curves (or optimal paths) in a Riemannian space manifold.

### Geometrized unit system

**geometrizedgeometric unitsgeometrized units**

Path curvature is the reciprocal of the magnitude of the curvature vector of a curve, so in geometric units it has the dimension of inverse length.

### Rindler coordinates

**Rindler spaceRindler horizonaccelerated coordinates**

One way to see this is to observe that the magnitude of the acceleration vector is just the path curvature of the corresponding world line.

### Elasticity of cell membranes

**cell membranesHelfrich’s law**

where k_n, k_g, and \tau_g are normal curvature, geodesic curvature, and geodesic torsion of the boundary curve, respectively.

### Four-vertex theorem

**Four Vertex Theorem**

The stereographic projection from the sphere to the plane preserves critical points of geodesic curvature.

### Capstan equation

**Belt friction equationEuler-Eytelwein**

where k_gis a geodesic curvature of the rope curve, k is a curvature of a rope curve, \mu_\tauis a coefficient of friction in the tangential direction.

### Horocycle

**horocyclic**

If the metric is normalized to have Gaussian curvature −1, then the horocycle is a curve of geodesic curvature 1 at every point.