An open cylindrical tube of radius and height is built by gluing a flat rectangular…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Topology and global structure

Invariance, Ch Integrating forms: Stokes's theorem, holes and Maxwell's equations

Problem

An open cylindrical tube of radius

a

and height

h

is built by gluing a flat rectangular strip along one pair of opposite edges with no twist, so that points

(0,z)

and

(2\pi a, z)

of the flat strip are identified for each

z\in[0,h]

. Someone argues: "A Möbius strip is made by gluing the ends of a strip after a half twist, and it has no consistent normal. Gluing a strip into a loop always identifies the two ends of the same normal field, so the untwisted cylinder must also fail to have a consistent normal, and no outward flux through it can be defined." ((a)) Decide whether the cylindrical tube is orientable, and if it is, write down an explicit, nowhere-zero, continuous outward unit normal field on it. ((b)) Identify the error in the argument quoted above: what property of the gluing actually decides orientability, and how does the cylinder differ from the Möbius strip on that count?

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