JacobiFields

# Jacobi Fields

A smooth tangent vector field $J\colon [0,1] \to T\mathcal M$ along a geodesic $g(\cdot;x,y)$ is called Jacobi field if it fulfills the ODE

$\displaystyle 0 = \frac{D}{dt}J + R(J,\dot g)\dot g,$

where $R$ is the Riemannian curvature tensor. Such Jacobi fields can be used to derive closed forms for the exponential map, the logarithmic map and the geodesic, all of them with respect to both arguments: Let $F\colon\mathcal N \to \mathcal M$ be given (for the $\exp_x\cdot$ we have $\mathcal N = T_x\mathcal M$, otherwise $\mathcal N=\mathcal M$) and denote by $\Xi_1,\ldots,\Xi_d$ an orthonormal frame along $g(\cdot;x,y)$ that diagonalizes the curvature tensor with corresponding eigenvalues $\kappa_1,\ldots,\kappa_d$. Note that on symmetric manifolds such a frame always exists.

Then $DF(x)[\eta] = \sum_{k=1}^d \langle \eta,\Xi_k(0)\rangle_x\beta(\kappa_k)\Xi_k(T)$ holds, where $T$ also depends on the function $F$ as the weights $\beta$. The values stem from solving the corresponding system of (decoupled) ODEs.

Note that in different references some factors might be a little different, for example when using unit speed geodesics.

The following weights functions are available

βDgx(κ,t,d)

weights for the jacobiField corresponding to the differential of the geodesic with respect to its start point $D_x g(t;x,y)[\eta]$. They are

$\beta(\kappa) = \begin{cases} \frac{\sinh(d(1-t)\sqrt{-\kappa})}{\sinh(d\sqrt{-\kappa})} &\text{ if }\kappa < 0,\\ 1-t & \text{ if } \kappa = 0,\\ \frac{\sin((1-t)d\sqrt{\kappa})}{\sinh(d\sqrt{\kappa})} &\text{ if }\kappa > 0. \end{cases}$

Due to a symmetry agrument, these are also used to compute $D_y g(t; x,y)[\eta]$

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βDexpx(κ,t,d)

weights for the jacobiField corresponding to the differential of the geodesic with respect to its start point $D_x \exp_x(\xi)[\eta]$. They are

$\beta(\kappa) = \begin{cases} \cosh(\sqrt{-\kappa})&\text{ if }\kappa < 0,\\ 1 & \text{ if } \kappa = 0,\\ \cos(\sqrt{\kappa}) &\text{ if }\kappa > 0. \end{cases}$

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βDexpξ(κ,t,d)

weights for the jacobiField corresponding to the differential of the geodesic with respect to its start point $D_\xi \exp_x(\xi)[\eta]$. They are

$\beta(\kappa) = \begin{cases} \frac{\sinh(d\sqrt{-\kappa})}{d\sqrt{-\kappa}}&\text{ if }\kappa < 0,\\ 1 & \text{ if } \kappa = 0,\\ \frac{\sin(d\sqrt{\kappa})}{\sqrt{d\kappa}}&\text{ if }\kappa > 0. \end{cases}$

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βDlogx(κ,t,d)

weights for thejacobiField corresponding to the differential of the geodesic with respect to its start point $D_x \log_xy[\eta]$. They are

$\beta(\kappa) = \begin{cases} -\sqrt{-\kappa}d\frac{\cosh(d\sqrt{-\kappa})}{\sinh(d\sqrt{-\kappa})}&\text{ if }\kappa < 0,\\ -1 & \text{ if } \kappa = 0,\\ -\sqrt{\kappa}d\frac{\cos(d\sqrt{\kappa})}{\sin(d\sqrt{\kappa})}&\text{ if }\kappa > 0. \end{cases}$

βDlogy(κ,t,d)
weights for the JacobiField corresponding to the differential of the logarithmic map with respect to its argument $D_y \log_xy[\eta]$. They are
$\beta(\kappa) = \begin{cases} \frac{ d\sqrt{-\kappa} }{\sinh(d\sqrt{-\kappa})}&\text{ if }\kappa < 0,\\ 1 & \text{ if } \kappa = 0,\\ \frac{ d\sqrt{\kappa} }{\sin(d\sqrt{\kappa})}&\text{ if }\kappa > 0. \end{cases}$