This function evaluates for with fixed the differenital
It is calculated a corresponding Jacobi field
Let , be an orthonormal basis in and its parallel transported frame, i.e. we have .
With the geodesic variation
and hence . We obtain which is also the derivtive of 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 solving the corresponding ODE and taking the derivative the coefficients depending on
where is the length of the geodesic. and the differential in total reads
1 2 3 4 5 6 7 8 9 10 % 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