# Charts of a torus (and other manifolds)

by mcafej
Tags: charts, manifolds, torus
 P: 350 A chart is a map that you use to navigate your way through some area. For example, the USGS makes a ton of maps so that you can look at the appropriate chart to check out the local terrain/landmarks. Usually, charts are marked to show latitude and longitude. These are coordinates that tell you where you are on that chart. An atlas is a book of charts that covers all the places of interest. The meaning of chart/atlas in manifold theory is the same in spirit. A chart is a map of the area near a point and the points on the charts have rectangular coordinates that tell you where you are on the chart. Given a manifold there are an infiinte number of charts. Just like on the globe there are an infinite number of possible charts because a chart is determined by the exact area I want to map and also by the manner in which I want to assign coordinates to the points on the chart (You dont' have to use latitude/longitude, you could make up another way to assign coordinates). Formally speaking, a chart consists of an open set U in the manifold, and a bijective mapping $\varphi: U\rightarrow \mathbb{R}^n .$ The image of the mapping is your flat "map" of the region U. The mapping you described does give a chart of the torus though to be precise you need to specify the domain and range so that you have a 1-1 mapping. The fact that you need more than 1 chart is a reflection of the fact that a single flat map cannot smoothly cover the torus unless you wrap it on itself.