Jacobi 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. 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 one parameter family of geodesics $$gamma_ au with $$gamma_0=gamma, then :$$J(t)=frac{partialgamma_ au(t)}{partial au}|_{ au=0}is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic $$gamma.

A vector field "J" along a geodesic $$gamma is said to be a Jacobi field if it satisfies the Jacobi equation::$$frac{D^2}{dt^2}J(t)+R(J(t),dotgamma(t))dotgamma(t)=0,where "D" denotes the covariant derivative with respect to the Levi-Civita connection, "R" the Riemann curvature tensor, and $$dotgamma(t)=dgamma(t)/dt and "t" is the parameter of the geodesic.On a complete Riemannian manifold, for any Jacobi field there is a family of geodesics $$gamma_ au describing the field (as in the preceding paragraph).

The Jacobi equation is a linear, second order ordinary differential equation;in particular, values of $$J and $$frac{D}{dt}J at one point of $$gamma uniquely determine the Jacobi field. Furthermore, the set of Jacobi fields along a given geodesic forms a real vector space of dimension twice the dimension of the manifold.

As trivial examples of Jacobi fields one can consider $$dotgamma(t) and $$tdotgamma(t). These correspond respectively to the following families of reparametrisations: $$gamma_ au(t)=gamma( au+t) and $$gamma_ au(t)=gamma((1+ au)t).

Any Jacobi field $$J can be represented in a unique way as a sum $$T+I, where $$T=adotgamma(t)+btdotgamma(t) is a linear combination of trivial Jacobi fields and $$I(t) is orthogonal to $$dotgamma(t), for all $$t. The field $$I then corresponds to the same variation of geodesics as $$J, only with changed parameterizations.

Motivating example

On a sphere, the geodesics through the North pole are great circles. Consider two such geodesics $$gamma_0 and $$gamma_ au with natural parameter, $$tin [0,pi] , separated by an angle $$au. The geodesic distance $$d(gamma_0(t),gamma_ au(t)) is :$$d(gamma_0(t),gamma_ au(t))=sin^{-1}igg(sin tsin ausqrt{1+cos^2 t an^2( au/2)}igg). Computing this requires knowing the geodesics. The most interesting information is just that:$$d(gamma_0(pi),gamma_ au(pi))=0, for any $$au. Instead, we can consider the derivative with respect to $$au at $$au=0::$$frac{partial}{partial au}igg|_{ au=0}d(gamma_0(t),gamma_ au(t))=|J(t)|=sin t. Notice that we still detect the intersection of the geodesics at $$t=pi. Notice further that to calculate this derivative we do not actually need to know $$d(gamma_0(t),gamma_ au(t)), rather, all we need do is solve the equation $$y"+y=0, for some given initial data.

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

olving the Jacobi equation

Let $$e_1(0)=dotgamma(0)/|dotgamma(0)| and complete this to get an orthonormal basis $$ig{e_i(0)ig} at $$T_{gamma(0)}M. Parallel transport it to get a basis $${e_i(t)} all along $$gamma. This gives an orthonormal basis with $$e_1(t)=dotgamma(t)/|dotgamma(t)|. The Jacobi field can be written in co-ordinates in terms of this basis as $$J(t)=y^k(t)e_k(t) and thus:$$frac{D}{dt}J=sum_kfrac{dy^k}{dt}e_k(t),quadfrac{D^2}{dt^2}J=sum_kfrac{d^2y^k}{dt^2}e_k(t), and the Jacobi equation can be rewritten as a system:$$frac{d^2y^k}{dt^2}+|dotgamma|^2sum_j y^j(t)langle R(e_j(t),e_1(t))e_1(t),e_k(t) angle=0 for each $$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 $${y^k}'(0), for all $$k.

Examples

Consider a geodesic $$gamma(t) with parallel orthonormal frame $$e_i(t), $$e_1(t)=dotgamma(t)/|dotgamma|, constructed as above.
* The vector fields along $$gamma given by $$dot gamma(t) and $$tdot gamma(t) are Jacobi fields.
* 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 $$dotgamma(t), $$tdotgamma(t) and $$exp(pm kt)e_i(t), where $$i>1.
*For Riemannian manifolds of constant positive curvature $$k^2, any Jacobi field is a linear combination of $$dotgamma(t), $$tdotgamma(t), $$sin(kt)e_i(t) and $$cos(kt)e_i(t), where $$i>1.
*The restriction of a Killing vector field to a geodesic is a Jacobi field in any Riemannian manifold.

ee also

* conjugate points
* N-Jacobi field

References

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