Commentary 2.2 Covering by a single chart

I was thinking about "Exercise 2.01 One chart for infinite cylinder". I could not understand why the Physics Forums solution by andrewkirk which invokes an annulus did not suffer from the same problem as a circle S¹. How could you map an infinite annulus to one chart, when the angle round the annulus must be measured and is 0 ≤ θ < 2π. So the coordinate θ cannot be mapped onto an open chart in R¹.

This is a similar question to asking why the plane in polar coordinates can be mapped on to one R² chart. Or the reverse that the plane in polar coordinates can not be mapped on to one R² chart. That last sentence is obviously not true.

R² in polar coordinates

The polar coordinates are r ≥ 0

and  θ where  0 ≤ θ < 2π

Neither of the coordinates are open on R¹.

The mapping of polar coordinates to R² is well known:

x = r cos θ, y = r sin θ.

So there is a mapping from the plane in polar coordinates to R². ✓

Conclusion

Coordinates on a manifold may not map onto charts in R¹ but this does not stop the manifold mapping onto a chart.

Therefore it may be safe to use the infinite open annulus to construct our chart for the infinite cylinder.

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