In this article we’ll see what a manifold is and how they are used. The way I like to think of manifolds is that they contain objects that aren’t necessarily numbers. A consequence of this is that we are free to put pretty much anything in a manifold; we can put numbers (of course) but we can also use a manifold to describe geometrical objects. However, a manifold only becomes useful in a mathematical sense when we use it in conjunction with a method for assigning numbers to the abstract objects. Traditionally this is done by way of an atlas in much the same way as an atlas of the Earth assigns co-ordinates (numbers) to points on the surface of the Earth (an abstract geometrical thing). An atlas is a collection of charts whose job it is to do the actual number assigning except that they do it a little bit at a time (doing it all at once can cause logical problems such as dealing with poles). This naturally leads to the question: where does the chart get it’s number from? How does it know to assign the right number to the right point on the manifold?
A chart is a function so it’s job is to eat a thing and spit out another thing, in this case it eats a point (or an abstract object) on a manifold and spits out a co-ordinate (a number). We decree the chart gets it’s numbers from the space of real numbers (or simply Euclidean space) one of the most structure-heavy spaces ever created by mathematicians, and as a result we find that it is extraordinarily overweight with rules and assumptions, an otherwise perfect place for our chart function to retrieve numbers from. How does a chart know to get the coordinate from Euclidean space? We’re talking about manifolds right? Not flat space.