The differential of the logarithmic map base point

This function evaluates for $F(x)=\log_xy$ with fixed $y\in\M$ the differenital $D_xF(x)[\eta].$

It is calculated a corresponding Jacobi field JacoiField.

Let $\xi_k$, $k=1,\ldots,d$ be an orthonormal basis in $T_x\M$ and $\Xi_k(t)\in T_{\geo{x}{y}(t)}\M$ its parallel transported frame, i.e. we have $\Xi(0) = \xi_k$.

With the geodesic variation

and hence $\geoS(t) = \geo{x}{y}(t) = \Gamma(0,t)$. We obtain $D_xF(x)[\xi_k] = \tfrac{D}{\partial t}\tfrac{D}{\partial s}\Gamma(s,t)\big|_{s=t=0}$ which is also the derivtive $J_k'(0)$ of the Jacobi field and $J_k(t) = \tfrac{D}{\partial s}\Gamma(s,t)\big|_{s=0}$ with boundary conditions $J_k(0) = \xi_k$ and $J(1) = 0$.

For $\xi_k$ being an orthonormal basis diagonalizing the curvature tensor on a symmetric Riemannian manifold (see Jacobi field) we obatin solving the corresponding ODE and taking the derivative the coefficients depending on $\kappa$

where $d_\geoS = D_\M(x,y)$ is the length of the geodesic. and the differential in total reads

Matlab Documentation

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% nu = AdjDxExp(x,xi,eta) - Derivative of Exp w.r.t. base point x
%    INPUT
%      x   : base point of the exponential
%      xi  : direction of the exponential
%     eta  : (in TxM) direction to take the derivative at.
%
%    OUTPUT
%     nu   : ( in TExp(x,xi)M ) - the DxExp with respect to eta
% ---
% MVIRT R. Bergmann, 2017-12-04

See also

• The Jacobi field
• The adjoint differential of the logairthmic map base point