Jacobi Fields

A smooth tangent vector field $J: [0,1] → T\mathcal M$ along a geodesic $g(⋅;x,y)$ 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 $F:\mathcal N → \mathcal M$ be given (for the $\exp_x⋅$ we have $\mathcal N = T_x\mathcal M$, otherwise $\mathcal N=\mathcal M$) and denote by $ξ_1,…,ξ_d$ an orthonormal frame along $g(⋅;x,y)$ that diagonalizes the curvature tensor with corresponding eigenvalues $κ_1,…,κ_d$. Note that on symmetric manifolds such a frame always exists.

Then $DF(x)[η] = \sum_{k=1}^d \langle η,ξ_k(0)\rangle_xβ(κ_k)ξ_k(T)$ holds, where $T$ also depends on the function $F$ as the weights $β$. 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

Manopt.adjoint_Jacobi_fieldFunction
Y = adjoint_Jacobi_field(M, p, q, t, X, β)
adjoint_Jacobi_field!(M, Y, p, q, t, X, β)

Compute the AdjointJacobiField $J$ along the geodesic $γ_{p,q}$ on the manifold $\mathcal M$ with initial conditions (depending on the application) $X ∈ T_{γ_{p,q}(t)}\mathcal M$ and weights $β$. The result is a vector $Y ∈ T_p\mathcal M$. The main difference to jacobi_field is the, that the input X and the output Y switched tangent spaces. The computation can be done in place of Y.

For detais see jacobi_field

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Manopt.jacobi_fieldFunction
Y = jacobi_field(M, p, q, t, X, β)
jacobi_field!(M, Y, p, q, t, X, β)

compute the Jacobi jield $J$ along the geodesic $γ_{p,q}$ on the manifold $\mathcal M$ with initial conditions (depending on the application) $X ∈ T_p\mathcal M$ and weights $β$. The result is a tangent vector Y from $T_{γ_{p,q}(t)}\mathcal M$. The computation can be done in place of Y.

See also

adjoint_Jacobi_field

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Manopt.βdifferential_exp_argumentMethod
βdifferential_exp_argument(κ,t,d)

weights for the jacobi_field corresponding to the differential of the geodesic with respect to its start point $D_X \exp_p X[Y]$. They are

\[β(κ) = \begin{cases} \frac{\sinh(d\sqrt{-κ})}{d\sqrt{-κ}}&\text{ if }κ < 0,\\ 1 & \text{ if } κ = 0,\\ \frac{\sin(d\sqrt{κ})}{\sqrt{dκ}}&\text{ if }κ > 0. \end{cases}\]

See also

differential_exp_argument, jacobi_field

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Manopt.βdifferential_geodesic_startpointMethod
βdifferential_geodesic_startpoint(κ,t,d)

weights for the jacobi_field corresponding to the differential of the geodesic with respect to its start point $D_x g(t;p,q)[X]$. They are

\[β(κ) = \begin{cases} \frac{\sinh(d(1-t)\sqrt{-κ})}{\sinh(d\sqrt{-κ})} &\text{ if }κ < 0,\\ 1-t & \text{ if } κ = 0,\\ \frac{\sin((1-t)d\sqrt{κ})}{\sinh(d\sqrt{κ})} &\text{ if }κ > 0. \end{cases}\]

Due to a symmetry agrument, these are also used to compute $D_q g(t; p,q)[η]$

See also

differential_geodesic_endpoint, differential_geodesic_startpoint, jacobi_field

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Manopt.βdifferential_log_argumentMethod
βdifferential_log_argument(κ,t,d)

weights for the JacobiField corresponding to the differential of the logarithmic map with respect to its argument $D_q \log_p q[X]$. They are

\[β(κ) = \begin{cases} \frac{ d\sqrt{-κ} }{\sinh(d\sqrt{-κ})}&\text{ if }κ < 0,\\ 1 & \text{ if } κ = 0,\\ \frac{ d\sqrt{κ} }{\sin(d\sqrt{κ})}&\text{ if }κ > 0. \end{cases}\]

See also

differential_log_basepoint, jacobi_field

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Manopt.βdifferential_log_basepointMethod
βdifferential_log_basepoint(κ,t,d)

weights for the jacobi_field corresponding to the differential of the geodesic with respect to its start point $D_p \log_p q[X]$. They are

\[β(κ) = \begin{cases} -\sqrt{-κ}d\frac{\cosh(d\sqrt{-κ})}{\sinh(d\sqrt{-κ})}&\text{ if }κ < 0,\\ -1 & \text{ if } κ = 0,\\ -\sqrt{κ}d\frac{\cos(d\sqrt{κ})}{\sin(d\sqrt{κ})}&\text{ if }κ > 0. \end{cases}\]

See also

differential_log_argument, differential_log_argument, jacobi_field

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