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You don’t need a metric for this, but I think you need to know at least what kind of connection your manifold is endowed with, for this to be possible at all (someone please correct me if I’m wrong). Given a connection, you can use geodesic deviation - you analyse all the possible ways that geodesics can deviate in the most general case (no Killing fields) on your manifold. From this you can deduce the number of functionally independent components of the Riemann tensor and torsion tensor, which straightforwardly gives you the dimensionality of the manifold. This is independent of any metric; it even works if there’s no metric at all on the manifold. Practically doing it may, however, be pretty cumbersome. Anyone know of an easier way?

The climate scientists told me to remind Bob Cross that "weather" and "climate" are two different things, and that ignorance of the science is no excuse.