Our Universe as a Manifold

Mateo Gomez

This research project gave a rigorous introduction to understanding spaces that locally look just like our familiar space, but globally may be curved. The perfect example of this is Earth: locally it looks flat, but globally it is a sphere. In fact, our very universe itself is assumed to have this local/global property, but being four-dimensional — three dimensions of space and one of time — we cannot “see” this as we can with Earth. Such spaces are called manifolds. The key question is: how does one do calculus on manifolds? How does one study their geometry, their curvature? Everything from the ground up has to be re-formulated: differentiability, vectors, the gradient, the Jacobian of a function, etc. All this needs to be understood before one can even begin to talk about geometry, or curvature, or gravity, or the universe and Einstein’s equations.