Crazy Mathman

Flying on the surface!

Posted on May 17, 2015 by Hitesh Gakhar

Remember the time when people thought that earth was flat? Nah, you probably don’t! Earth, or an approximated sphere/ellipsoid/ball sits inside $\mathbb{R}^3$ . However, to local residents of our puny planet, it might look like a plane. The above math-fiction-mumbo-jumbo I mentioned begs me to define something rigorous.

Definition 1: Let $X$ and $Y$ be topological spaces. If there is a map $f$ from $X$ to $Y$ such that $f$ is a bijective, continuous and open map, then we say that such a $f$ is a homeomorphism and $X$ and $Y$ are said to be homeomorphic to each other.

Definition 2: Let $X$ and $Y$ be topological spaces. If for all $x \in X$ , there is an open neighbourhood $U$ of $x$ such that $f(U)$ is open and $U$ is homeomorphic to $f(U)$ , then $f$ is called a local homeomorphism.

Now, if we take our usual sphere, its surface is locally homeomorphic to $\mathbb{R}^2$ . This is equivalent to saying that to a person on a large sphere(or our planet), the surface will look like a plane. To see the local homeomorphism, choose any point, take a small open disc on the surface around it. That will look like a lense. Flattening it out will give you a planar disc(Note that the map is injective, surjective onto the image, continuous and inverse continuous).

Now what would have happened if Earth was sitting in $\mathbb{R}^4$ and with an extra coordinate. To be precise, let us say the equation is

$x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1$ .

Let us choose a point $y = (y_1 , y_2 , y_3 , y_4)$ on the surface of our new sphere. Take a small disc $D$ . of radius $r$ centered at $y$ . Now, if we “flatten” $D$ in a similar way, what we get is a solid sphere or a 3 dimensional disc. This means that our new sphere is locally homeomorphic to $\mathbb{R}^3$ .

Therefore, walking on the surface of a 3-dimensional(as a manifold) sphere is the same as flying in $\mathbb{R}^3$ .