In General relativity, the metric tensor that satisfies Einstein's equations induces the Levi-Civita connection of that metric.
It is said that this connection is somehow "compatible" with the metric.
Technically Im told this means that straight lines (according to the connection) coincide with geodesics (according to the metric). However, this seems like an arbitrarily restrictive assumption from a mathematical point of view. Shouldn't it be possible for a single specific metric manifold with connection to have straight lines that are not necessarily geodesics? Why not?
So my main question is: what does this notion of "compatibility" of metric and connection really mean intuitively? Does it mean there cannot exist a metric manifold with connection whose metric and connection are incompatible? (I.e. is it a necessary condition?). Why is the intuitiv enotion of "compatibilty" captured formally by the "straight lines = geodesics" criterium?