Jacobi field - AbsoluteAstronomy.com

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

, a Jacobi field is a vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

along a geodesic

Geodesic

In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

in a Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space of all geodesics. They are named after Carl Jacobi.

Definitions and properties

Jacobi fields can be obtained in the following way: Take a smooth

Smooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

one parameter family of geodesics

with

, then

is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a
given geodesic

.

A vector field J along a geodesic

is said to be a Jacobi field if it satisfies the Jacobi equation:

where D denotes the covariant derivative

Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

with respect to the Levi-Civita connection

Levi-Civita connection

In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...

, R the Riemann curvature tensor

Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...

the tangent vector field, and t is the parameter of the geodesic.
On a complete

Complete space

In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....

Riemannian manifold, for any Jacobi field there is a family of geodesics

describing the field (as in the preceding paragraph).

The Jacobi equation is a linear

Linear differential equation

Linear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...

, second order ordinary

Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....

differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

;
in particular, values of

and

at one point of

uniquely determine the Jacobi field. Furthermore, the set of Jacobi fields along a given geodesic forms a real vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

of dimension twice the dimension of the manifold.

As trivial examples of Jacobi fields one can consider

and

. These correspond respectively to the following families of reparametrisations:

and

.

Any Jacobi field

can be represented in a unique way as a sum

, where

is a linear combination of trivial Jacobi fields and

is orthogonal to

, for all

.
The field

then corresponds to the same variation of geodesics as

, only with changed parameterizations.

Motivating example

On a sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

, the geodesic

Geodesic

s through the North pole are great circle

Great circle

A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center...

s. Consider two such geodesics

and

with natural parameter,

, separated by an angle

. The geodesic distance

Computing this requires knowing the geodesics. The most interesting information is just that

, for any

.
Instead, we can consider the derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

with respect to

Notice that we still detect the intersection

Intersection (set theory)

In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

of the geodesics at

. Notice further that to calculate this derivative we do not actually need to know

,
rather, all we need do is solve the equation

,
for some given initial data.

Jacobi fields give a natural generalization of this phenomenon to arbitrary Riemannian manifold

Riemannian manifold

Solving the Jacobi equation

Let

and complete this to get an orthonormal basis

. Parallel transport

Parallel transport

In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...

it to get a basis

all along

.
This gives an orthonormal basis with

. The Jacobi field can be written in co-ordinates in terms of this basis as

and thus

and the Jacobi equation can be rewritten as a system

for each

. This way we get a linear ordinary differential equation (ODE).
Since this ODE has smooth

Smooth function

coefficient

Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

s we have that solutions exist for all

and are unique, given

and

, for all

Examples

Consider a geodesic

with parallel orthonormal frame

, constructed as above.

The vector fields along given by and are Jacobi fields.
In Euclidean space (as well as for spaces of constant zero sectional curvature
Sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...

) Jacobi fields are simply those fields linear in .
For Riemannian manifolds of constant negative sectional curvature , any Jacobi field is a linear combination of , and , where .
For Riemannian manifolds of constant positive sectional curvature , any Jacobi field is a linear combination of , , and , where .
The restriction of a Killing vector field
Killing vector field
In mathematics, a Killing vector field , named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold...

to a geodesic is a Jacobi field in any Riemannian manifold.
The Jacobi fields correspond to the geodesics on the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

(with respect to the metric on induced by the metric on ).

Definitions and properties

Motivating example

Solving the Jacobi equation

Examples

See also