# 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.