Finding geodesics joining given points

Abstract Finding a geodesic joining two given points in complete path-connected Riemannian manifold requires much more effort than determining from initial data. This is because it harder to solve boundary value problems problems. Shooting methods attempt by solving sequence of problems, and usually need good guess succeed. The present paper finds $$\gamma :[0,1]\rightarrow M$$ ? : [ 0 , 1 ] ? M on the M with ? (0) = x 0 (1) 1 dividing interval [0,1] into several sub-intervals, preferably just enough enable for problem each subinterval. Then consecutive endpoints (local junctions) found single shooting. Our algorithm then adjusts junctions, either minimizing total squared norm differences between associated velocities using gradient descent, or (2) nonlinear system equations Newton’s method. compared known leapfrog numerical experiments 2-dimensional ellipsoid Ell(2) left-invariant 3-dimensional special orthogonal group S O (3). We find method converges faster when junctions are needed, that can be starting descent (1).