Metrics on affine connections

Question

In one of the paper's I read this statement: "the affine geodesics of the Cartan connections (group geodesics) are metric-free". What does this really mean? paper: http://www-sop.inria.fr/asclepios/Publications/Xavier.Pennec/Lorenzi_Pennec_IJCV2012.pdf

Answer 1

Here is the full quote:

"As in finite dimension, the affine geodesics of the Cartan connections (group geodesics) are metric-free (the Hilbert metric is only used to specify the space on which is modeled the Lie Algebra) and generally differ from the Riemannian geodesics of LDDMM."

In other words, the affine geodesics of the Cartan connections (in this setup) generally differ from the Riemannian geodesics. That is, the affine geodesics of the Cartan connections (in this setup) are not length-minimizing for the metric. I think that's the point.

See, for example, the last line of page 3:

"Given a general affine connection, there may not exist any Riemannian metric for which affine geodesics are length minimizing."