This function evaluates for with fixed the differenital
It is calculated a corresponding Jacobi field
Note that the tangent space is its own tangent space, i.e. .
Let , be an orthonormal basis in and its parallel transported frame, i.e. we have .
With the geodesic variation
We obtain which is also the Jacobi field and with boundary conditions and .
For being an orthonormal basis diagonalizing the curvature tensor on a symmetric Riemannian manifold (see Jacobi field) we obatin slving the corresponding ODE depending on the coefficients
where is the length of the geodesic. and the differential in total reads
1 2 3 4 5 6 7 8 9 10 % nu = DxExp(x,xi,eta) - Derivative of Exp w.r.t. xi % INPUT % x : base point of the exponential % xi : direction of the exponential % eta : (in Txi(TxM)=TxM direction to take the derivative at. % % OUTPUT % nu : ( in TExp(x,xi)M) ) - the adjoint of DxExp with respect to eta % --- % MVIRT R. Bergmann, 2017-12-04