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Jacobi field

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In Riemannian geometry, a Jacobi field is a vector field along a geodesic γ {\displaystyle \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 γ τ {\displaystyle \gamma _{\tau }} with γ 0 = γ {\displaystyle \gamma _{0}=\gamma } , then

J ( t ) = ∂ γ τ ( t ) ∂ τ | τ = 0 {\displaystyle 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 γ {\displaystyle \gamma } .

A vector field J along a geodesic γ {\displaystyle \gamma } is said to be a Jacobi field if it satisfies the Jacobi equation:

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,}

where D denotes the covariant derivative with respect to the Levi-Civita connection, R the Riemann curvature tensor, γ ˙ ( t ) = d γ ( t ) / d t {\displaystyle {\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 γ τ {\displaystyle \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 {\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 vector space of dimension twice the dimension of the manifold.

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 reparametrisations: 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)} .

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 parameterizations.

Motivating example

On a unit sphere, the geodesics through the North pole are great circles. Consider two such geodesics γ 0 {\displaystyle \gamma _{0}} and γ τ {\displaystyle \gamma _{\tau }} with natural parameter, t ∈ [ 0 , π ] {\displaystyle t\in } , separated by an angle τ {\displaystyle \tau } . The geodesic distance

d ( γ 0 ( t ) , γ τ ( t ) ) {\displaystyle d(\gamma _{0}(t),\gamma _{\tau }(t))\,}

is

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 )}.}

Computing this requires knowing the geodesics. The most interesting information is just that

d ( γ 0 ( π ) , γ τ ( π ) ) = 0 {\displaystyle d(\gamma _{0}(\pi ),\gamma _{\tau }(\pi ))=0\,} , for any τ {\displaystyle \tau } .

Instead, we can consider the derivative with respect to τ {\displaystyle \tau } at τ = 0 {\displaystyle \tau =0} :

∂ ∂ 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.}

Notice that we still detect the intersection of the geodesics at t = π {\displaystyle t=\pi } . Notice further that to calculate this derivative we do not actually need to know

d ( γ 0 ( t ) , γ τ ( t ) ) {\displaystyle d(\gamma _{0}(t),\gamma _{\tau }(t))\,} ,

rather, all we need do is solve the equation

y ″ + y = 0 {\displaystyle 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 ) = γ ˙ ( 0 ) / | γ ˙ ( 0 ) | {\displaystyle e_{1}(0)={\dot {\gamma }}(0)/|{\dot {\gamma }}(0)|} and complete this to get an orthonormal basis { e i ( 0 ) } {\displaystyle {\big \{}e_{i}(0){\big \}}} at T γ ( 0 ) M {\displaystyle T_{\gamma (0)}M} . Parallel transport 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

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),}

and the Jacobi equation can be rewritten as a system

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}

for each k {\displaystyle 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 {\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} .

Examples

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.

  • 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.
  • In Euclidean space (as well as for spaces of constant zero sectional curvature) Jacobi fields are simply those fields linear in t {\displaystyle t} .
  • 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} .
  • 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} .
  • The restriction of a Killing vector field to a geodesic is a Jacobi field in any Riemannian manifold.

See also

References

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