
# The adjoint differential of the end point of a geodesic

This function evaluates for $F(y)=\geo{x}{y}(t)$ with fixed $x\in\M$ and $t\in\mathbb R$ the adjoint differenital $D^*_yF(y)[\eta].$

It is calculated a corresponding adjoint Jacobi field AdjJacoiField. Since we can rewrite the problem as computing the differential of the reverted geodesic $\geo{y}{x}(1-t)$, this differential is computed using the adjoint differential of the start point of a geodesic.

### Matlab Documentation

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
%
%    For a function f: M \mapsto R and fixed x,t we have for the
%    gradient of g(y) = 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(x,y,t)) direction to take the Adjoint derivative at.
%
%    OUTPUT
%     xi   : ( in TyM ) - the adjoint of DyGeo with respect to eta
% ---
% MVIRT R. Bergmann, 2017-12-04