Main Question or Discussion Point
My question is, is it forbidden to have a connection not compatible with the metric?
As @Orodruin has said, that doesn't help.Senseless, not allowed
To expand on that, it is perfectly possible to have a connection without your manifold having a metric at all so quite clearly metric compatibility cannot be a constraint on a general connection. There are many possible meanings of ”forbidden” or ”senseless” depending on the context of the use of those words. Hence my request for specification.The answer to your question is no, connections dont always have metric compatibility.