Jacobi field

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In Riemannian geometry, a Jacobi field is a certain type of vector field along a geodesic [\gamma] in a Riemannian manifold. Jacobi fields are one of the basic objects of study in Riemannian geometry; for the origin of the name, see Carl Jacobi.

Contents

1 Definitions and properties
2 Motivating example
3 Solving the Jacobi equation
4 Examples
5 References

Definitions and properties

Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics [\gamma_\tau] with [\gamma_0=\gamma], then

[J(t)=\partial\gamma_\tau(t)/\partial \tau|_]

is a Jacobi field.

A field J is a Jacobi field if and only if it satisfies the Jacobi equation:

[\fracJ(t)+R(J(t),\dot\gamma(t))\dot\gamma(t)=0,]

where D denotes the Levi-Civita connection, R the curvature tensor and [\dot\gamma(t)=d\gamma(t)/dt]. 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).

The Jacobi equation is a linear second order ordinary differential equation; in particular, values of [J] and [\fracJ] at one point of [\gamma] define uniquely the Jacobi field. Further, the sum of Jacobi fields on a given geodesic is again a Jacobi field.

As trivial examples of Jacobi fields one can consider [\dot\gamma(t)] and [t\dot\gamma(t)]. These correspond respectively to the following families of reparametrisations: [\gamma_\tau(t)=\gamma(\tau+t)] and [\gamma_\tau(t)=\gamma((1+\tau)t)].

Any Jacobi field field [J] can be represented in a unique way as a sum [T+I], where [T=a\dot\gamma(t)+bt\dot\gamma(t)] is a linear combination of trivial Jacobi fields and [I(t)] is orthogonal to [\dot\gamma(t)], for all [t]. The field [I] then corresponds to the same variation of geodesics as [J], only with changed parametrizations.

Motivating example

On a sphere, the geodesics through the North pole are great circles. Consider two such geodesics [\gamma_0] and [\gamma_\tau] with natural parameter, [t\in [0,pi]], separated by an angle [\tau]. The geodesic distance [d(\gamma_0(t),\gamma_\tau(t))] is

[d(\gamma_0(t),\gamma_\tau(t))=\sin^\bigg(\sin t\sin\tau\sqrt\bigg).]

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

[d(\gamma_0(\pi),\gamma_\tau(\pi))=0], for any [\tau].

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

[\frac\bigg|_d(\gamma_0(t),\gamma_\tau(t))=|J(t)|=\sin t.]

Notice that we still detect the intersection of the geodesics at [t=\pi]. Notice further that to calculate this derivative we do not actually need to know [d(\gamma_0(t),\gamma_\tau(t))], rather, all we need do is solve the equation [y''+y=0], for some given initial data.

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

Solving the Jacobi equation

Let [e_1(0)=\dot\gamma(0)/|\dot\gamma(0)|] and complete this to get an orthonormal basis [\big\] at [T_M]. Parallel transport it to get a basis [\] all along [\gamma]. This gives an orthonormal basis with [e_1(t)=\dot\gamma(t)/|\dot\gamma(t)|]. The Jacobi field is [J(t)=y^k(t)e_k(t)] and thus

[\fracJ=\sum_k\frace_k(t),\quad\fracJ=\sum_k\frace_k(t),]

and the Jacobi equation can be rewritten as a system

[\frac+|\dot\gamma|^2\sum_j y^j(t)\langle R(e_j(t),e_1(t))e_1(t),e_k(t)\rangle=0]

for each [k]. This way we get a linear ordinary differential equation (ODE). Since this ODE has smooth coefficients we have that solutions exist for all [t] and are unique, given [y^k(0)] and ['(0)], for all [k].

Examples

Consider a geodesic [\gamma(t)] with parallel basis frame [e_i(t)], [e_1(t)=\dot\gamma(t)/|\dot\gamma|], constructed as above.

In Euclidean space (as well as for spaces of constant zero curvature) Jacobi fields are simply those fields linear in [t].

For Riemannian manifolds of constant negative curvature [-k^2], any Jacobi field is a linear combination of [\dot\gamma(t)], [t\dot\gamma(t)] and [\exp(\pm kt)e_i(t)], where [i>1].

For Riemannian manifolds of constant positive curvature [k^2], any Jacobi field is a linear combination of [\dot\gamma(t)], [t\dot\gamma(t)], [\sin(kt)e_i(t)] and [\cos(kt)e_i(t)], where [i>1].

References

[do Carmo] M. P. do Carmo, Riemannian Geometry, Universitext, 1992.

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