A little preamble: an affine connection on the tangent bundle of a Riemannian (or pseudo-Riemannian) manifold is compatible with the metric if the covariant derivative of the metric is zero. Of course, there is the unique Levi-Civita connection which is compatible with the metric and also torsion-free, but in general a metric-compatible connection may have torsion.
Now, a geodesic of the connection is a curve whose tangent vector parallel transports itself. If the connection is the Levi-Civita connection, a geodesic is also a curve of extremal path length between two points. Here's my question: is there a name for connections whose geodesics also minimize path lengths? (i.e., whose geodesics are the same as those for the Levi-Civita connection?)
I've figured out that such connections need not be compatible with the metric, though they must have torsion (except for the Levi-Civita connection itself), and they can also be metric-compatible while having torsion. In the latter case the torsion tensor turns out to be fully antisymmetric: $$T_{ijk}=T_{[ijk]}$$ In 3 dimensions, parallel transport by a connection with this kind of torsion causes vectors to rotate around the path's tangent vector, which is neat.
Do these connections have any other interesting properties?