Incompatible coordinate charts on the two dimensional sphere

Question

Most of the examples of the charts for $S^2$ involve compatible ones like:

1. Stereographic projections from the North and South Pole.

2. Projections onto the disk bound by the equator.

What would be simplest example of two coordinate charts for $S^2$ that are NOT compatible with each other?

Answer 1

Let $(x, y)$ denote Cartesian coordinates on the plane. Smooth homeomorphisms such as \begin{align*} \phi_{0}(x, y) &= (x, y), \\ \phi_{1}(x, y) &= (x^{3}, y), \\ \phi_{2}(x, y) &= (x, y^{3}), \\ \phi_{3}(x, y) &= (x^{3}, y^{3}), \end{align*} among an infinite-dimensional space's worth of other examples, define pairwise-incompatible charts on the plane.

To get charts on the sphere, it suffices to pick two of these and follow them by a compatible pair of charts for the sphere, such as stereographic projection from the north pole and stereographic projection from the south pole.

The conceptual points are:

• All mappings in question are homeomorphisms of the plane to itself, or from a punctured sphere to the plane;
• Incompatible charts composed with compatible charts are incompatible. (Contrapositively: Compatible charts composed with compatible charts are compatible.)

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One doesn't normally see this type of example mentioned explicitly: It's typical behavior for charts of topological manifolds (homeomorphisms are "rarely" diffeomorphisms), and inapplicable to the study of smooth manifolds (no smooth atlas contains two incompatible charts).