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.
Contents

Definitions and properties 1

Motivating example 2

Solving the Jacobi equation 3

Examples 4

See also 5

References 6
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)=\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 with respect to the LeviCivita connection, R the 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 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 \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 \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 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 parameterizations.
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^{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 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 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\{e_i(0)\big\} 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)=\dot\gamma(t)/\dot\gamma(t). The Jacobi field can be written in coordinates 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 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)=\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) 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 to a geodesic is a Jacobi field in any Riemannian manifold.

The Jacobi fields correspond to the geodesics on the tangent bundle (with respect to the metric on TM induced by the metric on M).
See also
References

[do Carmo] M. P. do Carmo, Riemannian Geometry, Universitext, 1992.
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.