Flying on the surface!

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 .