Complex sentence: Here the manifold "M" is only equipped with a torsion-free affine connection and is said to be affine symmetric, if its geodesic symmetries are all globally defined and affine maps.
Split into simpler sentences: Here the manifold "M" is only equipped with a torsion-free affine connection. In this case "M" is said to be affine symmetric, if its geodesic symmetries are all globally defined and affine maps.