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Main Question or Discussion Point
I'm looking for a simple example of a differentiable manifold that doesn't have an associated riemann metric.
thanks
thanks
We have to be careful with definitions here. In particular, it seems to me that a non-second-countable smooth manifold may be equipped with a local Riemann metric but not be a metric space.The proof that every manifold has a metric (as well as the proof of Whitney's embedding theorem) relies on paracompactness. If you drop this requirement, you can have all sorts of aberrations.
In fact, if your space has a metric, it has to be second countable (delta-balls type argument).
True, true. The space must be path-connected for what I said to hold.We have to be careful with definitions here. In particular, it seems to me that a non-second-countable smooth manifold may be equipped with a local Riemann metric but not be a metric space.
Not enough. If it's non-second-countable, the integral that lets us go from local Riemannian metric to global metricity may diverge.True, true. The space must be path-connected for what I said to hold.