math lessons - Jacobi field

In Riemannian geometry, a Jacobi field is a certain type of vector field along a geodesic $γ$ in a Riemannian manifold. Jacobi fields are one of the basic objects of study in Riemannian geometry; for the origin of the name, see Carl Jacobi.

Contents

1 Definitions and properties

2 Motivating example

3 Solving the Jacobi Equation

4 Examples

5 References

Definitions and properties

Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics $γ τ$ with $γ 0 = γ$ , then

$J(t)=\partial\gamma_\tau(t)/\partial \tau|_{\tau=0}$

is a Jacobi field.

A field J is a Jacobi field if and only 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 Levi-Civita connection, R the curvature tensor and $\dot\gamma(t)=d\gamma(t)/dt$ . On a complete Riemannian manifold, for any Jacobi field there is a family of geodesics $γ τ$ 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 $γ$ define uniquely the Jacobi field. Further, the sum of Jacobi fields on a given geodesic is again a Jacobi field.

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: $γ τ (t) = γ(τ + t)$ and $γ τ (t) = γ((1 + τ) t)$ .

Any Jacobi field 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 parametrizations.

Motivating example

On a sphere, the geodesics through the North pole are great circles. Consider two such geodesics $γ 0$ and $γ τ$ with natural parameter, $t\in [0,\pi]$ , separated by an angle $τ$ . The geodesic distance $d (γ 0 (t),γ τ (t))$ 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 (γ 0 (π),γ τ (π)) = 0

, for any

τ

Instead, we can consider the derivative with respect to $τ$ at $τ = 0$ :

$\frac{\partial}{\partial\tau}\bigg|_{\tau=0}d(\gamma_0(t),\gamma_\tau(t))=|J(t)|=\sin t.$

Notice that we still detect the intersection of the geodesics at $t = π$ . Notice further that to calculate this derivative we do not actually need to know $d (γ 0 (t),γ τ (t))$ , 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 $T γ(0) M$ . Parallel transport it to get a basis ${e i (t)}$ all along $γ$ . This gives an orthonormal basis with $e_1(t)=\dot\gamma(t)/|\dot\gamma(t)|$ . The Jacobi field is $J (t) = y k (t) e k (t)$ 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 $γ(t)$ with parallel basis frame $e i (t)$ , $e_1(t)=\dot\gamma(t)/|\dot\gamma|$ , constructed as above.

In Euclidean space (as well as for spaces of constant zero curvature) Jacobi fields are simply those fields linear in $t$ .

For Riemannian manifolds of constant negative 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 curvature $k 2$ , any Jacobi field is a linear combination of $\dot\gamma(t)$ , $t\dot\gamma(t)$ , $sin(k t) e i (t)$ and $cos(k t) e i (t)$ , where $i > 1$ .

References

[do Carmo] M. P. do Carmo, Riemannian Geometry, Universitext, 1992.

Mathematics Encyclopedia and Lessons

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Jacobi field

Definitions and properties

Motivating example

Solving the Jacobi Equation

Examples

References

Mathematics Encyclopedia and Lessons

Lessons

Popular Subjects

References

Jacobi field

Definitions and properties

Motivating example

Solving the Jacobi Equation

Examples

References

Popular
Subjects