# 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.

### Sectional curvature

**curvaturecurvature tensorsmanifolds with constant sectional curvature**