# Jacobi field

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In [Riemannian geometry](/source/Riemannian_geometry), a **Jacobi field** is a [vector field](/source/Vector_field) along a [geodesic](/source/Geodesic) \gamma in a [Riemannian manifold](/source/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](/source/Tangent_space) to the geodesic in the space of all geodesics. They are named after [Carl Jacobi](/source/Carl_Gustav_Jacob_Jacobi).

## Definitions and properties

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

- J(t)=\left.\frac{\partial\gamma_\tau(t)}{\partial \tau}\right|_{\tau=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),\dot\gamma(t))\dot\gamma(t)=0,

where *D* denotes the [covariant derivative](/source/Covariant_derivative) with respect to the [Levi-Civita connection](/source/Levi-Civita_connection), *R* the [Riemann curvature tensor](/source/Riemann_curvature_tensor), \dot\gamma(t)=d\gamma(t)/dt the tangent vector field, and *t* is the parameter of the geodesic. On a [complete](/source/Complete_space) 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](/source/Linear_differential_equation), second order [ordinary differential equation](/source/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](/source/Vector_space) of dimension twice the dimension of the manifold.

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 reparametrizations: \gamma_\tau(t)=\gamma(\tau+t) and \gamma_\tau(t)=\gamma((1+\tau)t).

Any Jacobi 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 [unit sphere](/source/Unit_sphere), the [geodesics](/source/Geodesic) through the North pole are [great circles](/source/Great_circle). 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^{-1}\bigg(\sin t\sin\tau\sqrt{1+\cos^2 t\tan^2(\tau/2)}\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](/source/Derivative) with respect to \tau at \tau=0:

- \frac{\partial}{\partial\tau}\bigg|_{\tau=0}d(\gamma_0(t),\gamma_\tau(t))=|J(t)|=\sin t.

Notice that we still detect the [intersection](/source/Intersection_(set_theory)) 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](/source/Riemannian_manifold).

## Solving the Jacobi equation

Let e_1(0)=\dot\gamma(0)/|\dot\gamma(0)| and complete this to get an [orthonormal](/source/Orthonormal) basis \big\{e_i(0)\big\} at T_{\gamma(0)}M. [Parallel transport](/source/Parallel_transport) it to get a basis \{e_i(t)\} all along \gamma. This gives an [orthonormal basis](/source/Orthonormal_basis) with e_1(t)=\dot\gamma(t)/|\dot\gamma(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_k\frac{dy^k}{dt}e_k(t),\quad\frac{D^2}{dt^2}J=\sum_k\frac{d^2y^k}{dt^2}e_k(t),

and the Jacobi equation can be rewritten as a system

- \frac{d^2y^k}{dt^2}+|\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](/source/Smooth_function) [coefficients](/source/Coefficient) 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)=\dot\gamma(t)/|\dot\gamma|, constructed as above.

- The vector fields along \gamma given by \dot \gamma(t) and t\dot \gamma(t) are Jacobi fields.
- In Euclidean space (as well as for spaces of constant zero [sectional curvature](/source/Sectional_curvature)) Jacobi fields are simply those fields linear in t.
- For Riemannian manifolds of constant negative sectional 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 sectional 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.
- The restriction of a [Killing vector field](/source/Killing_vector_field) to a geodesic is a Jacobi field in any Riemannian manifold.

## See also

- [Conjugate points](/source/Conjugate_points)
- [Geodesic deviation equation](/source/Geodesic_deviation_equation)
- [Rauch comparison theorem](/source/Rauch_comparison_theorem)
- [N-Jacobi field](/source/N-Jacobi_field)

## References

- [Manfredo Perdigão do Carmo](/source/Manfredo_do_Carmo). Riemannian geometry. Translated from the second Portuguese edition by Francis Flaherty. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992. xiv+300 pp. ISBN 0-8176-3490-8
- [Jeff Cheeger](/source/Jeff_Cheeger) and [David G. Ebin](/source/David_Gregory_Ebin). Comparison theorems in Riemannian geometry. Revised reprint of the 1975 original. AMS Chelsea Publishing, Providence, RI, 2008. x+168 pp. ISBN 978-0-8218-4417-5
- [Shoshichi Kobayashi](/source/Shoshichi_Kobayashi) and [Katsumi Nomizu](/source/Katsumi_Nomizu). [Foundations of differential geometry](/source/Foundations_of_differential_geometry). Vol. II. Reprint of the 1969 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996. xvi+468 pp. ISBN 0-471-15732-5
- [Barrett O'Neill](/source/Barrett_O'Neill). Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. xiii+468 pp. ISBN 0-12-526740-1

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