# Jacobi field

Jacobi equation
In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic.wikipedia
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### Geodesic

geodesicsgeodesic flowgeodesic equation
In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. On a sphere, the geodesics through the North pole are great circles.
In an appropriate sense, zeros of the second variation along a geodesic γ arise along Jacobi fields.

### Riemann curvature tensor

Riemann tensorcurvature tensorcurvature
:where D denotes the covariant derivative with respect to the Levi-Civita connection, R the Riemann curvature tensor, the tangent vector field, and t is the parameter of the geodesic.
The curvature tensor represents the tidal force experienced by a rigid body moving along a geodesic in a sense made precise by the Jacobi equation.

### Geodesic deviation

Geodesic deviation equation
In differential geometry, the geodesic deviation equation is more commonly known as the Jacobi equation.

### Conjugate points

conjugateConjugate point
Then p and q are conjugate points along \gamma if there exists a non-zero Jacobi field along \gamma that vanishes at p and q.

### Rauch comparison theorem

This theorem is formulated using Jacobi fields to measure the variation in geodesics.

### Riemannian geometry

Riemannianlocal to global theoremsRiemann geometry
In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic.

### Vector field

In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic.

### Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic.

### Carl Gustav Jacob Jacobi

JacobiCarl Gustav JacobiCarl Jacobi
They are named after Carl Jacobi.

### Smoothness

smoothsmooth functionsmooth map
Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics \gamma_\tau with, then Since this ODE has smooth coefficients we have that solutions exist for all t and are unique, given y^k(0) and {y^k}'(0), for all k.

### Covariant derivative

covariant differentiationtensor derivativecovariant differential
:where D denotes the covariant derivative with respect to the Levi-Civita connection, R the Riemann curvature tensor, the tangent vector field, and t is the parameter of the geodesic.

### Levi-Civita connection

Christoffel symbolconnectionsLevi-Civita
:where D denotes the covariant derivative with respect to the Levi-Civita connection, R the Riemann curvature tensor, the tangent vector field, and t is the parameter of the geodesic.

### Complete metric space

completecompletioncompleteness
On a complete Riemannian manifold, for any Jacobi field there is a family of geodesics \gamma_\tau describing the field (as in the preceding paragraph).

### Linear differential equation

constant coefficientslinearlinear differential equations
The Jacobi equation is a linear, second order ordinary differential equation;

### Ordinary differential equation

ordinary differential equationsordinaryODE
The Jacobi equation is a linear, second order ordinary differential equation;

### Differential equation

differential equationsdifferentialsecond-order differential equation
The Jacobi equation is a linear, second order ordinary differential equation;

### Vector space

vectorvector spacesvectors
Furthermore, the set of Jacobi fields along a given geodesic forms a real vector space of dimension twice the dimension of the manifold.

### Sphere

sphericalhemisphereglobose
On a sphere, the geodesics through the North pole are great circles.

### Great circle

Great Circle Routegreat circlesarcs of great circle
On a sphere, the geodesics through the North pole are great circles.

### Derivative

differentiationdifferentiablefirst derivative
Instead, we can consider the derivative with respect to \tau at \tau=0:

### Intersection (set theory)

intersectionintersectionsset intersection
: Notice that we still detect the intersection of the geodesics at t=\pi.

### Orthonormality

orthonormalorthonormal setorthonormal sequence
Let and complete this to get an orthonormal basis at.

### Parallel transport

parallelparallel-transporttransported
Parallel transport it to get a basis \{e_i(t)\} all along \gamma.