Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Jacobi field
open-in-new
<h2 id="definitions-and-properties">Definitions and properties</h2> <p>Jacobi fields can be obtained in the following way: Take a <a href="/facts/Smooth_function/mffL4ch2">smooth</a> one parameter family of geodesics γ τ {\displaystyle \gamma _{\tau }} with γ 0 = γ {\displaystyle \gamma _{0}=\gamma } , then </p> J ( t ) = ∂ γ τ ( t ) ∂ τ | τ = 0 {\displaystyle J(t)=\left.{\frac {\partial \gamma _{\tau }(t)}{\partial \tau }}\right|_{\tau =0}} <p>is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic γ {\displaystyle \gamma } . </p><p>A vector field <i>J</i> along a geodesic γ {\displaystyle \gamma } is said to be a Jacobi field if it satisfies the Jacobi equation: </p> D 2 d t 2 J ( t ) + R ( J ( t ) , γ ˙ ( t ) ) γ ˙ ( t ) = 0 , {\displaystyle {\frac {D^{2}}{dt^{2}}}J(t)+R(J(t),{\dot {\gamma }}(t)){\dot {\gamma }}(t)=0,} <p>where <i>D</i> denotes the <a href="/facts/Covariant_derivative/sRGN0Xh6">covariant derivative</a> with respect to the <a href="/facts/Levi-Civita_connection/16KGSQUY">Levi-Civita connection</a>, <i>R</i> the <a href="/facts/Riemann_curvature_tensor/KyeHCWEh">Riemann curvature tensor</a>, γ ˙ ( t ) = d γ ( t ) / d t {\displaystyle {\dot {\gamma }}(t)=d\gamma (t)/dt} the tangent vector field, and <i>t</i> is the parameter of the geodesic. On a <a href="/facts/Complete_space/lsckmU8G">complete</a> Riemannian manifold, for any Jacobi field there is a family of geodesics γ τ {\displaystyle \gamma _{\tau }} describing the field (as in the preceding paragraph). </p><p>The Jacobi equation is a <a href="/facts/Linear_differential_equation/nLEbIIVf">linear</a>, second order <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equation</a>; in particular, values of J {\displaystyle J} and D d t J {\displaystyle {\frac {D}{dt}}J} at one point of γ {\displaystyle \gamma } uniquely determine the Jacobi field. Furthermore, the set of Jacobi fields along a given geodesic forms a real <a href="/facts/Vector_space/M1pxMLx2">vector space</a> of dimension twice the dimension of the manifold. </p><p>As trivial examples of Jacobi fields one can consider γ ˙ ( t ) {\displaystyle {\dot {\gamma }}(t)} and t γ ˙ ( t ) {\displaystyle t{\dot {\gamma }}(t)} . These correspond respectively to the following families of reparametrizations: γ τ ( t ) = γ ( τ + t ) {\displaystyle \gamma _{\tau }(t)=\gamma (\tau +t)} and γ τ ( t ) = γ ( ( 1 + τ ) t ) {\displaystyle \gamma _{\tau }(t)=\gamma ((1+\tau )t)} . </p><p>Any Jacobi field J {\displaystyle J} can be represented in a unique way as a sum T + I {\displaystyle T+I} , where T = a γ ˙ ( t ) + b t γ ˙ ( t ) {\displaystyle T=a{\dot {\gamma }}(t)+bt{\dot {\gamma }}(t)} is a linear combination of trivial Jacobi fields and I ( t ) {\displaystyle I(t)} is orthogonal to γ ˙ ( t ) {\displaystyle {\dot {\gamma }}(t)} , for all t {\displaystyle t} . The field I {\displaystyle I} then corresponds to the same variation of geodesics as J {\displaystyle J} , only with changed parametrizations. </p> <h2 id="motivating-example">Motivating example</h2> <p>On a <a href="/facts/Unit_sphere/5UeoTcug">unit sphere</a>, the <a href="/facts/Geodesic/fbWR6l80">geodesics</a> through the North pole are <a href="/facts/Great_circle/x4fq6eNm">great circles</a>. Consider two such geodesics γ 0 {\displaystyle \gamma _{0}} and γ τ {\displaystyle \gamma _{\tau }} with natural parameter, t ∈ [ 0 , π ] {\displaystyle t\in [0,\pi ]} , separated by an angle τ {\displaystyle \tau } . The geodesic distance </p> d ( γ 0 ( t ) , γ τ ( t ) ) {\displaystyle d(\gamma _{0}(t),\gamma _{\tau }(t))\,} <p>is </p> d ( γ 0 ( t ) , γ τ ( t ) ) = sin − 1 ( sin t sin τ 1 + cos 2 t tan 2 ( τ / 2 ) ) . {\displaystyle d(\gamma _{0}(t),\gamma _{\tau }(t))=\sin ^{-1}{\bigg (}\sin t\sin \tau {\sqrt {1+\cos ^{2}t\tan ^{2}(\tau /2)}}{\bigg )}.} <p>Computing this requires knowing the geodesics. The most interesting information is just that </p> d ( γ 0 ( π ) , γ τ ( π ) ) = 0 {\displaystyle d(\gamma _{0}(\pi ),\gamma _{\tau }(\pi ))=0\,} , for any τ {\displaystyle \tau } . <p>Instead, we can consider the <a href="/facts/Derivative/7Ito70Dh">derivative</a> with respect to τ {\displaystyle \tau } at τ = 0 {\displaystyle \tau =0} : </p> ∂ ∂ τ | τ = 0 d ( γ 0 ( t ) , γ τ ( t ) ) = | J ( t ) | = sin t . {\displaystyle {\frac {\partial }{\partial \tau }}{\bigg |}_{\tau =0}d(\gamma _{0}(t),\gamma _{\tau }(t))=|J(t)|=\sin t.} <p>Notice that we still detect the <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersection</a> of the geodesics at t = π {\displaystyle t=\pi } . Notice further that to calculate this derivative we do not actually need to know </p> d ( γ 0 ( t ) , γ τ ( t ) ) {\displaystyle d(\gamma _{0}(t),\gamma _{\tau }(t))\,} , <p>rather, all we need do is solve the equation </p> y ″ + y = 0 {\displaystyle y''+y=0\,} , <p>for some given initial data. </p><p>Jacobi fields give a natural generalization of this phenomenon to arbitrary <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifolds</a>. </p> <h2 id="solving-the-jacobi-equation">Solving the Jacobi equation</h2> <p>Let e 1 ( 0 ) = γ ˙ ( 0 ) / | γ ˙ ( 0 ) | {\displaystyle e_{1}(0)={\dot {\gamma }}(0)/|{\dot {\gamma }}(0)|} and complete this to get an <a href="/facts/Orthonormal/JfgL1nAh">orthonormal</a> basis { e i ( 0 ) } {\displaystyle {\big \{}e_{i}(0){\big \}}} at T γ ( 0 ) M {\displaystyle T_{\gamma (0)}M} . <a href="/facts/Parallel_transport/QX0Wfcte">Parallel transport</a> it to get a basis { e i ( t ) } {\displaystyle \{e_{i}(t)\}} all along γ {\displaystyle \gamma } . This gives an orthonormal basis with e 1 ( t ) = γ ˙ ( t ) / | γ ˙ ( t ) | {\displaystyle 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 ) {\displaystyle J(t)=y^{k}(t)e_{k}(t)} and thus </p> D d t J = ∑ k d y k d t e k ( t ) , D 2 d t 2 J = ∑ k d 2 y k d t 2 e k ( t ) , {\displaystyle {\frac {D}{dt}}J=\sum _{k}{\frac {dy^{k}}{dt}}e_{k}(t),\quad {\frac {D^{2}}{dt^{2}}}J=\sum _{k}{\frac {d^{2}y^{k}}{dt^{2}}}e_{k}(t),} <p>and the Jacobi equation can be rewritten as a system </p> d 2 y k d t 2 + | γ ˙ | 2 ∑ j y j ( t ) ⟨ R ( e j ( t ) , e 1 ( t ) ) e 1 ( t ) , e k ( t ) ⟩ = 0 {\displaystyle {\frac {d^{2}y^{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} <p>for each k {\displaystyle k} . This way we get a linear ordinary differential equation (ODE). Since this ODE has <a href="/facts/Smooth_function/mffL4ch2">smooth</a> <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a> we have that solutions exist for all t {\displaystyle t} and are unique, given y k ( 0 ) {\displaystyle y^{k}(0)} and y k ′ ( 0 ) {\displaystyle {y^{k}}'(0)} , for all k {\displaystyle k} . </p> <h2 id="examples">Examples</h2> <p>Consider a geodesic γ ( t ) {\displaystyle \gamma (t)} with parallel orthonormal frame e i ( t ) {\displaystyle e_{i}(t)} , e 1 ( t ) = γ ˙ ( t ) / | γ ˙ | {\displaystyle e_{1}(t)={\dot {\gamma }}(t)/|{\dot {\gamma }}|} , constructed as above. </p> <ul><li>The vector fields along γ {\displaystyle \gamma } given by γ ˙ ( t ) {\displaystyle {\dot {\gamma }}(t)} and t γ ˙ ( t ) {\displaystyle t{\dot {\gamma }}(t)} are Jacobi fields.</li> <li>In Euclidean space (as well as for spaces of constant zero <a href="/facts/Sectional_curvature/GpI9zQDh">sectional curvature</a>) Jacobi fields are simply those fields linear in t {\displaystyle t} .</li> <li>For Riemannian manifolds of constant negative sectional curvature − k 2 {\displaystyle -k^{2}} , any Jacobi field is a linear combination of γ ˙ ( t ) {\displaystyle {\dot {\gamma }}(t)} , t γ ˙ ( t ) {\displaystyle t{\dot {\gamma }}(t)} and exp ( ± k t ) e i ( t ) {\displaystyle \exp(\pm kt)e_{i}(t)} , where i > 1 {\displaystyle i>1} .</li> <li>For Riemannian manifolds of constant positive sectional curvature k 2 {\displaystyle k^{2}} , any Jacobi field is a linear combination of γ ˙ ( t ) {\displaystyle {\dot {\gamma }}(t)} , t γ ˙ ( t ) {\displaystyle t{\dot {\gamma }}(t)} , sin ( k t ) e i ( t ) {\displaystyle \sin(kt)e_{i}(t)} and cos ( k t ) e i ( t ) {\displaystyle \cos(kt)e_{i}(t)} , where i > 1 {\displaystyle i>1} .</li> <li>The restriction of a <a href="/facts/Killing_vector_field/Ane9xIdV">Killing vector field</a> to a geodesic is a Jacobi field in any Riemannian manifold.</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Conjugate_points/7GbGNFxF">Conjugate points</a></li> <li><a href="/facts/Geodesic_deviation_equation/4Y3gk4eq">Geodesic deviation equation</a></li> <li><a href="/facts/Rauch_comparison_theorem/IlWdVJDq">Rauch comparison theorem</a></li> <li>N-Jacobi field</li></ul> <ul><li><a href="/facts/Manfredo_do_Carmo/5yxcMKQB">Manfredo Perdigão do Carmo</a>. Riemannian geometry. Translated from the second Portuguese edition by Francis Flaherty. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992. xiv+300 pp. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8176-3490-8</li> <li><a href="/facts/Jeff_Cheeger/VTmcprIR">Jeff Cheeger</a> and <a href="/facts/David_Gregory_Ebin/Gk6jDSQG">David G. Ebin</a>. Comparison theorems in Riemannian geometry. Revised reprint of the 1975 original. AMS Chelsea Publishing, Providence, RI, 2008. x+168 pp. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-4417-5</li> <li><a href="/facts/Shoshichi_Kobayashi/P3rHgfGs">Shoshichi Kobayashi</a> and <a href="/facts/Katsumi_Nomizu/Xhs8rgXm">Katsumi Nomizu</a>. 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. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-15732-5</li> <li><a href="/facts/Barrett_O%27Neill/mnbSvRIV">Barrett O'Neill</a>. 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. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-526740-1</li></ul>