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

vector fieldsvectorgradient flow
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.

Coefficient

coefficientsleading coefficientfactor
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.