An Intoduction to Manifolds
\tBefore we can understand Riemann Surfaces, we first have to look at manifolds. A manifold is a special kind of surface defined by a topological space which is considered “locally Euclidean”. This means that in small neighborhoods around each point on the manifold, the surface can be understood and expressed using regular geometry and vectors. A good example of this is Earth. We know that Earth is a sphere, but when we look at the neighborhood around us, it appears to be a flat plane. It is what allows us to make two dimensional maps of the Earth, even though we know that we live in three dimensional space. Manifolds can typically be understood through charts, which are graphs that express each of these locally Euclidean neighborhoods on the surface. The collection of these charts is known as the manifold’s atlas. Depending on how one looks at a manifold, it could produce a number of different atlasses, but each still defines the manifold completely through its charts.
\tThe image above shows an example of a manifold and two of the charts that define it. Each area around a specific point can be presented on a “regular” graph that we are used to using. Charts express the surface through pieces in a space that is one dimension lower than the manifold. This is like how we can use Stokes Theorem to solve integrals involving cones, cylinders, and spheres, by projecting them into two dimensional space and treating it like a circle. In the image, there is an area of overlap between the green and purple spaces that is colored in blue. The blue space is not the same for both charts, so a function known as a transition map. The transition map between two spaces, α and β, is expressed as the composition β ૦ α-1. This maps all points in the blue space of the green neighborhood to the points of the blue space of the purple neighborhood, showing that there are no holes or discontinuities in the surface depending on how and where it is viewed. Understanding manifolds is a necessary first step toward understanding and interpreting and manipulating Riemann Surfaces.
