Connection (mathematics)

In mathematics, "connection" generally refers to a way of connecting points or objects in a geometric space. This can be thought of like connecting dots on paper, but in a more complex and abstract way.

Imagine you have a bunch of dots on a piece of paper. You want to draw lines to connect them in a certain way. But you can't just draw random lines - they have to follow some rules. For example, you might want to draw lines that are as short as possible, or that make certain angles with each other.

In math, connections are like those rules for connecting dots. But instead of dots on paper, we might be talking about points in a three-dimensional space, or even more dimensions than that. The rules for connecting them depend on the specific situation we're working with.

One example of a connection is the "covariant derivative." This is a way of connecting points in a curved space, like the surface of a sphere. Because the surface is curved, the rules for connecting points have to be different than they would be in a flat space, like a sheet of paper. The covariant derivative takes into account the curvature of the surface and makes sure that lines connecting points follow the right rules.

Connections are used in many areas of math, including geometry, topology, and physics. They help us understand how different points or objects are related to each other in a space, and how we can move between them in specific ways.