Jacobi field

Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics ${\displaystyle \gamma _{\tau }}$ with ${\displaystyle \gamma _{0}=\gamma }$, then

${\displaystyle J(t)=\left.{\frac {\partial \gamma _{\tau }(t)}{\partial \tau }}\right|_{\tau =0}}$

is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic ${\displaystyle \gamma }$.

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

${\displaystyle {\frac {D^{2}}{dt^{2}}}J(t)+R(J(t),{\dot {\gamma }}(t)){\dot {\gamma }}(t)=0,}$

where D denotes the covariant derivative with respect to the Levi-Civita connection, R the Riemann curvature tensor, ${\displaystyle {\dot {\gamma }}(t)=d\gamma (t)/dt}$ 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 ${\displaystyle \gamma _{\tau }}$ describing the field (as in the preceding paragraph).

The Jacobi equation is a linear, second order ordinary differential equation; in particular, values of ${\displaystyle J}$ and ${\displaystyle {\frac {D}{dt}}J}$ at one point of ${\displaystyle \gamma }$ 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 ${\displaystyle {\dot {\gamma }}(t)}$ and ${\displaystyle t{\dot {\gamma }}(t)}$. These correspond respectively to the following families of reparametrisations: ${\displaystyle \gamma _{\tau }(t)=\gamma (\tau +t)}$ and ${\displaystyle \gamma _{\tau }(t)=\gamma ((1+\tau )t)}$.

Any Jacobi field ${\displaystyle J}$ can be represented in a unique way as a sum ${\displaystyle T+I}$, where ${\displaystyle T=a{\dot {\gamma }}(t)+bt{\dot {\gamma }}(t)}$ is a linear combination of trivial Jacobi fields and ${\displaystyle I(t)}$ is orthogonal to ${\displaystyle {\dot {\gamma }}(t)}$, for all ${\displaystyle t}$. The field ${\displaystyle I}$ then corresponds to the same variation of geodesics as ${\displaystyle J}$, only with changed parameterizations.

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

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

is

${\displaystyle d(\gamma _{0}(t),\gamma _{\tau }(t))=\sin ^{-1}{\bigg (}\sin t\sin \tau {\sqrt {1+\cos ^{2}t\tan ^{2}(\tau /2)}}{\bigg )}.}$

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

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

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

${\displaystyle {\frac {\partial }{\partial \tau }}{\bigg |}_{\tau =0}d(\gamma _{0}(t),\gamma _{\tau }(t))=|J(t)|=\sin t.}$

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

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

rather, all we need do is solve the equation

${\displaystyle y''+y=0\,}$,

for some given initial data.

Jacobi fields give a natural generalization of this phenomenon to arbitrary Riemannian manifolds.

Let ${\displaystyle e_{1}(0)={\dot {\gamma }}(0)/|{\dot {\gamma }}(0)|}$ and complete this to get an orthonormal basis ${\displaystyle {\big \{}e_{i}(0){\big \}}}$ at ${\displaystyle T_{\gamma (0)}M}$. Parallel transport it to get a basis ${\displaystyle \{e_{i}(t)\}}$ all along ${\displaystyle \gamma }$. This gives an orthonormal basis with ${\displaystyle e_{1}(t)={\dot {\gamma }}(t)/|{\dot {\gamma }}(t)|}$. The Jacobi field can be written in co-ordinates in terms of this basis as ${\displaystyle J(t)=y^{k}(t)e_{k}(t)}$ and thus

${\displaystyle {\frac {D}{dt}}J=\sum _{k}{\frac {dy^{k}}{dt}}e_{k}(t),\quad {\frac {D^{2}}{dt^{2}}}J=\sum _{k}{\frac {d^{2}y^{k}}{dt^{2}}}e_{k}(t),}$

and the Jacobi equation can be rewritten as a system

${\displaystyle {\frac {d^{2}y^{k}}{dt^{2}}}+|{\dot {\gamma }}|^{2}\sum _{j}y^{j}(t)\langle R(e_{j}(t),e_{1}(t))e_{1}(t),e_{k}(t)\rangle =0}$

for each ${\displaystyle k}$. This way we get a linear ordinary differential equation (ODE). Since this ODE has smooth coefficients we have that solutions exist for all ${\displaystyle t}$ and are unique, given ${\displaystyle y^{k}(0)}$ and ${\displaystyle {y^{k}}'(0)}$, for all ${\displaystyle k}$.

Consider a geodesic ${\displaystyle \gamma (t)}$ with parallel orthonormal frame ${\displaystyle e_{i}(t)}$, ${\displaystyle e_{1}(t)={\dot {\gamma }}(t)/|{\dot {\gamma }}|}$, constructed as above.

• The vector fields along ${\displaystyle \gamma }$ given by ${\displaystyle {\dot {\gamma }}(t)}$ and ${\displaystyle t{\dot {\gamma }}(t)}$ are Jacobi fields.
• In Euclidean space (as well as for spaces of constant zero sectional curvature) Jacobi fields are simply those fields linear in ${\displaystyle t}$.
• For Riemannian manifolds of constant negative sectional curvature ${\displaystyle -k^{2}}$, any Jacobi field is a linear combination of ${\displaystyle {\dot {\gamma }}(t)}$, ${\displaystyle t{\dot {\gamma }}(t)}$ and ${\displaystyle \exp(\pm kt)e_{i}(t)}$, where ${\displaystyle i>1}$.
• For Riemannian manifolds of constant positive sectional curvature ${\displaystyle k^{2}}$, any Jacobi field is a linear combination of ${\displaystyle {\dot {\gamma }}(t)}$, ${\displaystyle t{\dot {\gamma }}(t)}$, ${\displaystyle \sin(kt)e_{i}(t)}$ and ${\displaystyle \cos(kt)e_{i}(t)}$, where ${\displaystyle i>1}$.
• The restriction of a Killing vector field to a geodesic is a Jacobi field in any Riemannian manifold.

• Conjugate points
• Geodesic deviation equation
• Rauch comparison theorem
• N-Jacobi field

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