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 S

^{1}. 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 R^{1}.
This is a similar question to asking why the plane in polar coordinates
can be mapped on to one R

^{2}chart. Or the reverse that the plane in polar coordinates can not be mapped on to one R^{2}chart. That last sentence is obviously not true.R^{2} in polar coordinates |

The polar coordinates are r ≥ 0

and θ where 0 ≤ θ < 2π

Neither of the coordinates are open on R

^{1}.
The mapping of polar coordinates to R

^{2}is well known:
x = r cos θ, y = r sin θ.

So there is a mapping from the plane in polar coordinates to
R

^{2}. ✓#### Conclusion

**Coordinates on a manifold may not map onto charts in R**

^{1}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.

Available as pdf

Available as pdf

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