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| In [[Riemannian geometry]], a '''Jacobi field''' is a [[vector field]] along a [[geodesic]] <math>\gamma</math> 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 Gustav Jacob Jacobi|Carl Jacobi]].
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| ==Definitions and properties==
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| Jacobi fields can be obtained in the following way: Take a [[smooth function|smooth]] one parameter family of geodesics <math>\gamma_\tau</math> with <math>\gamma_0=\gamma</math>, then
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| :<math>J(t)=\left.\frac{\partial\gamma_\tau(t)}{\partial \tau}\right|_{\tau=0}</math>
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| is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a
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| given geodesic <math>\gamma</math>.
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| A vector field ''J'' along a geodesic <math>\gamma</math> is said to be a '''Jacobi field''' if it satisfies the '''Jacobi equation''':
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| :<math>\frac{D^2}{dt^2}J(t)+R(J(t),\dot\gamma(t))\dot\gamma(t)=0,</math>
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| where ''D'' denotes the [[covariant derivative]] with respect to the [[Levi-Civita connection]], ''R'' the [[Riemann curvature tensor]], <math>\dot\gamma(t)=d\gamma(t)/dt</math> the tangent vector field, and ''t'' is the parameter of the geodesic.
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| On a [[Complete space|complete]] Riemannian manifold, for any Jacobi field there is a family of geodesics <math>\gamma_\tau</math> describing the field (as in the preceding paragraph).
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| The Jacobi equation is a [[linear differential equation|linear]], second order [[ordinary differential equation|ordinary]] [[differential equation]];
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| in particular, values of <math>J</math> and <math>\frac{D}{dt}J</math> at one point of <math>\gamma</math> 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.
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| As trivial examples of Jacobi fields one can consider <math>\dot\gamma(t)</math> and <math>t\dot\gamma(t)</math>. These correspond respectively to the following families of reparametrisations: <math>\gamma_\tau(t)=\gamma(\tau+t)</math> and <math>\gamma_\tau(t)=\gamma((1+\tau)t)</math>.
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| Any Jacobi field <math>J</math> can be represented in a unique way as a sum <math>T+I</math>, where <math>T=a\dot\gamma(t)+bt\dot\gamma(t)</math> is a linear combination of trivial Jacobi fields and <math>I(t)</math> is orthogonal to <math>\dot\gamma(t)</math>, for all <math>t</math>.
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| The field <math>I</math> then corresponds to the same variation of geodesics as <math>J</math>, only with changed parameterizations.
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| ==Motivating example==
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| On a [[sphere]], the [[geodesic]]s through the North pole are [[great circle]]s. Consider two such geodesics <math>\gamma_0</math> and <math>\gamma_\tau</math> with natural parameter, <math>t\in [0,\pi]</math>, separated by an angle <math>\tau</math>. The geodesic distance
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| :<math>d(\gamma_0(t),\gamma_\tau(t)) \,</math> | |
| is
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| :<math>d(\gamma_0(t),\gamma_\tau(t))=\sin^{-1}\bigg(\sin t\sin\tau\sqrt{1+\cos^2 t\tan^2(\tau/2)}\bigg).</math>
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| Computing this requires knowing the geodesics. The most interesting information is just that
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| :<math>d(\gamma_0(\pi),\gamma_\tau(\pi))=0 \,</math>, for any <math>\tau</math>.
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| Instead, we can consider the [[derivative]] with respect to <math>\tau</math> at <math>\tau=0</math>:
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| :<math>\frac{\partial}{\partial\tau}\bigg|_{\tau=0}d(\gamma_0(t),\gamma_\tau(t))=|J(t)|=\sin t.</math>
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| Notice that we still detect the [[intersection (set theory)|intersection]] of the geodesics at <math>t=\pi</math>. Notice further that to calculate this derivative we do not actually need to know
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| :<math>d(\gamma_0(t),\gamma_\tau(t)) \,</math>, | |
| rather, all we need do is solve the equation
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| :<math>y''+y=0 \,</math>,
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| for some given initial data.
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| Jacobi fields give a natural generalization of this phenomenon to arbitrary [[Riemannian manifold]]s.
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| ==Solving the Jacobi equation==
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| Let <math>e_1(0)=\dot\gamma(0)/|\dot\gamma(0)|</math> and complete this to get an [[orthonormal]] basis <math>\big\{e_i(0)\big\}</math> at <math>T_{\gamma(0)}M</math>. [[Parallel transport]] it to get a basis <math>\{e_i(t)\}</math> all along <math>\gamma</math>.
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| This gives an orthonormal basis with <math>e_1(t)=\dot\gamma(t)/|\dot\gamma(t)|</math>. The Jacobi field can be written in co-ordinates in terms of this basis as <math>J(t)=y^k(t)e_k(t)</math> and thus
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| :<math>\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),</math>
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| and the Jacobi equation can be rewritten as a system
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| :<math>\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</math>
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| for each <math>k</math>. This way we get a linear ordinary differential equation (ODE). | |
| Since this ODE has [[smooth function|smooth]] [[coefficient]]s we have that solutions exist for all <math>t</math> and are unique, given <math>y^k(0)</math> and <math>{y^k}'(0)</math>, for all <math>k</math>.
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| ==Examples==
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| Consider a geodesic <math>\gamma(t)</math> with parallel orthonormal frame <math>e_i(t)</math>, <math>e_1(t)=\dot\gamma(t)/|\dot\gamma|</math>, constructed as above.
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| * The vector fields along <math>\gamma</math> given by <math>\dot \gamma(t)</math> and <math>t\dot \gamma(t)</math> are Jacobi fields.
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| * In Euclidean space (as well as for spaces of constant zero [[sectional curvature]]) Jacobi fields are simply those fields linear in <math>t</math>.
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| *For Riemannian manifolds of constant negative sectional curvature <math>-k^2</math>, any Jacobi field is a linear combination of <math>\dot\gamma(t)</math>, <math>t\dot\gamma(t)</math> and <math>\exp(\pm kt)e_i(t)</math>, where <math>i>1</math>.
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| *For Riemannian manifolds of constant positive sectional curvature <math>k^2</math>, any Jacobi field is a linear combination of <math>\dot\gamma(t)</math>, <math>t\dot\gamma(t)</math>, <math>\sin(kt)e_i(t)</math> and <math>\cos(kt)e_i(t)</math>, where <math>i>1</math>.
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| *The restriction of a [[Killing vector field]] to a geodesic is a Jacobi field in any Riemannian manifold.
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| *The Jacobi fields correspond to the geodesics on the [[tangent bundle]] (with respect to the metric on <math>TM</math> induced by the metric on <math>M</math>).
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| ==See also==
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| * [[conjugate points]]
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| * [[Geodesic deviation equation]]
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| * [[Rauch comparison theorem]]
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| * [[N-Jacobi field]]
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| ==References== | |
| *[do Carmo] M. P. do Carmo, ''Riemannian Geometry'', Universitext, 1992.
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| [[Category:Riemannian geometry]]
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| [[Category:Equations]]
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