Exercise 2.03 2D Torus is a manifold

Question

Show that the two-dimensional torus T² is a manifold, by explicitly constructing an appropriate atlas. (Not a maximal one, obviously).

Answer

Fig 1: A Torus, with help from Wikipedia

We can define a torus as points in R³ that lie on the surface created by rotating a small circle of radius r round a larger circle of radius R. The larger circle is in the X-Y plane. Any point on the torus can be given by (θ , κ) as shown in Fig 1. There are no limits on θ , κ.

I realised that I could map to infinite cylinders touching the inner and outer edges of the torus and from them to annuli as in Exercise 2.01. No doubt our cunning author had set this trap. Having been on that wild goose chase, I came to a linear map from T² to the annuli and realised that we can also map from T² to two open rectangles much as one can map from S¹ to R¹. Fig 2 shows a section at some θ through our torus. We almost have the two charts U_O (Outer) and U_I (Inner) we just need to cut them and unroll them to make open rectangles. We cut the outer U_O at θ = 0 and the inner U_I at θ = -π. If we cut them both at the same θ it would lead to disaster.

Fig 2: Section through torus and rectangular maps.

This is where it gets complicated and we end up with a few maps, so;

Fig 4: Showing the transformations. T² is exactly U_I  ∩ U_O. 

We now want to check that the charts obey the strict conditions, as given in the book before Fig 2.13 at the bottom of page 29, to make an atlas from T² to R². I have adapted them for the exercise.

1. "The union of the charts U_I  , U_O is equal to T²; that is, U_I  , U_O cover T²."

We have two charts U_I  , U_O; their union manifestly covers T² by construction. ✓

We could check that there are no cracks in the union by combining the limits on ϕ_I  , ϕ_O. We have  -π< θ<2π and -π<κ<2π. That covers T². ✓

2. "The charts are smoothly sewn together. More precisely, if two charts overlap, U_I ∩ U_O ≠∅, then the map (ϕ_I ∘ ϕ_O^-1) takes points in ϕ_O (U_I ∩ U_O  ) ⊂  R² onto* an open set ϕ_I (U_I ∩ U_O ) ⊂ R², and all these maps must be C^∞. The reverse for  (ϕ_O ∘ ϕ_I^-1) also applies.

* We remind ourselves that onto means each point of the target has at least one point of the source mapped onto it.

The first part was easy as shown. The second part was fiddly and you can read the detail and wild goose chase in Ex 2.03 2D Torus is a manifold.pdf. It has helped me greatly to understand manifolds.