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11.2 Circles, Spheres, and Hyperspheres

An $n$-sphere, written $S^{n}$, is defined as the set of $(n + 1)$-dimensional unit vectors, $$S^{n} = \{ x \in \mathbb{R}^{n+1} \, : \, \Vert x \Vert = 1 \}.$$

Even though $S^n$ is made up of points in $(n+1)$ dimensions, it is only an $n$-dimensional manifold. For example, $S^2$ is defined as a set of points in $\mathbb{R}^3$, but each such point may be described uniquely by a latitude and longitude. Geometrically, the surface defined by $S^2$ in $\mathbb{R}^3$ behaves locally like a plane, i.e., $\mathbb{R}^2$. However, the overall shape of $S^2$ is not like a plane in that is compact (i.e., there is a maximum distance between points). If you set off around the globe in a “straight line” (i.e., a geodesic), you wind up back where you started eventually; that is why the geodesics on the sphere ($S^2$) are called “great circles,” and why we need to use some clever representations to do circular or spherical statistics.

Even though $S^{n-1}$ behaves locally like $\mathbb{R}^{n-1}$, there is no way to smoothly map between them. For example, because latitude and longitude work on a modular basis (wrapping at $2\pi$ radians in natural units), they do not produce a smooth map.

Like a bounded interval $(a, b)$, in geometric terms, a sphere is compact in that the distance between any two points is bounded.