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My question is, is it forbidden to have a connection not compatible with the metric?
Senseless, not allowedWhat do you mean by ”forbidden”?
You are not helping yourself by not being more specific. You need to define the context of your question.Senseless, not allowed
Yes it is strictly forbidden. It is in the Geneva convention.My question is, is it forbidden to have a connection not compatible with the metric?
Name, rank (both co- and contra-variant parts), and serial number only, right?Yes it is strictly forbidden. It is in the Geneva convention.
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.