Wednesday, 15 August 2018

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.

Available as pdf

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