Example. The n-sphere [0094]

We can define the n-dimensional sphere as the set of vectors in with norm one. There are several ways to see that this defines a manifold.

1. Using charts. We can give charts by using stereographic projections. The stereographic projection is only well defined onto if we take out the pole. We can simply use two stereographic projections, one missing the northpole, and one missing the southpole.

2. As a local zero set of a regular height function. Consider the function . Notice that iff . Notice that the differential of is surjective on since it is given by . Its only zero is the origin, which does not lie in .

3. Locally as the graph of a smooth function. Consider the map given by . By restriciting the domain we obtain a smooth well defined map whose image is the upper hemisphere (without the equator). We can use to hit the lower hemisphere. Using coordinate permutations we can move a small-enough neighbourhood of any point into the upper or lower hemisphere.

TODO: Details