
# The differential of the start point of a geodesic

This function evaluates for $F(x)=\geo{x}{y}(t)$ with fixed $y\in\M$ and $t\in\mathbb R$ 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

we obtain $D_xF(x)[\xi_k] = \tfrac{D}{\partial s}\Gamma(s,t)\big|_{s=0}$ which is also 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 depending on $\kappa$ the coefficients

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

### Matlab Documentation

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
% xi = DxGeo(x,y,t,eta) - Compute the Derivative D_xGeo(t; x,y)[eta]
%    i.e. of geo(x,y,t) with respect to the start point x.
%
%    For a function f: M \mapsto R and fixed y,t we have for the
%    gradient of g(x) = f(geo(x,y,t)) that
%    hence with the Adjoint we obtain
%    This function hence only requires eta=grad f to computed
%    the chain rule.
%
%    INPUT
%      x   : start point of a geodesic, g(x,y,0)=x
%      y   : end point of a geodesic, geo(x,y,1) = y
%      t   : [0,1] a point on the geodesic to be evaluated,
%            may exceed [0,1] to leave the segment between x and y
%     eta  : (in Tg(t;x,y)M) direction to take the Adjoint derivative at.
%
%    OUTPUT
%     xi   : ( in TxM ) - the adjoint of DxGeo with respect to eta
% ---
% MVIRT R. Bergmann, 2017-12-04