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 Thanks for the reply but it does not directly relate to my question : I wrote specifically about a flat metric with only ONE additional metric function, as in the example I gave, and in Einstein's entwurf metric. Calculating the Riemann tensor for such a metric (for the conditions stated) shows that it indeed is curved, so that part is not my question. I am asking about the legitimacy of the 'reasoning', and then asking something similar regarding connection coefficients, again where there is only ONE, or only one which is 'large'. (I also would like to know whether a statement to this effect can be made for a non-metric connection) [BTW: The second metric you brought has something missing,dX?dx?]