Sphere in terms of manifolds

Question

Can we view a sphere as a two-dimensional manifold? Is this the reason why we call a sphere $S^2$?

Answer 1

The sphere $S^2$ is in a sense a two dimensional subset of $\mathbb R^3$, as all so called 'surfaces' are (although this is not a very precise term). It is actually a two dimensional (differentiable) manifold, since it is around each one of its points the image of a (diffeomorphic*) map from an open region of $\mathbb R^2$ (two variables, and hence 'two dimensional') to $S^2$. For instance, if we pick $(0,0,-1)\in S^2$, it is in the (open) 'southern hemisphere' of $S^2$, which is the image of $\phi \colon D \subset \mathbb R^2 \to S^2\subset\mathbb R^3$, $$\phi(x,y)=\left(x,y,-\sqrt{x^2+y^2}\right),$$ where $D$ is the unitary open disc $\{(x,y)\in\mathbb R^2\,\colon\, x^2+y^2<1\}$ (in particular $(0,0,-1)=\phi(0,0)$, and $(0,0)\in D$).

All this formalizes the intuitive notion that a 'piece' of the sphere $S^2$ (as happens with other 'surfaces') may be thought of as the result of deforming (not only continuously but also 'differentiably') a piece of plane, and hence it is two dimensional.

In the same fashion, the border of the disc $D$, that is the circumference $$\{(x,y)\in\mathbb R^2\,\colon\, x^2+y^2=1\},$$ can be considered as formed by 'arcs' which result from deforming segments of the real line; since a line has dimension one, the circumference (and any other 'nice' curve) is said to be one dimensional and that's why we use the notation $$S^1=\{(x,y)\in\mathbb R^2\,\colon\, x^2+y^2=1\}.$$

By analogy, $$S^0=\{x\in\mathbb R\,\colon\, x^2=1\}=\{-1,1\}$$ or $$S^3=\{(w,x,y,z)\in \mathbb R^4\,\colon\, w^2+x^2+y^2+z^2=1\}.$$ And in general, for $n\in \mathbb N$, $$S^n=\{(x_0,x_1,\ldots,x_n)\in \mathbb R^{n+1}\,\colon\, x_0^2+x_1^2+\cdots+x_n^2=1\}.$$

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• A map is diffeomorphic if it's bijective, differentiable and it's inverse is also differentiable. The precise meaning of differentiability of $\phi^{-1}$ in cases like the one mentioned here is made precise in the study of differential geometry.

Answer 2

Geometers call the surface of the usual sphere the 3-sphere, while topologists refer to it as the 2-sphere and denote it S^2

http://mathworld.wolfram.com/Sphere.html