- #1

Figaro

- 103

- 7

## Homework Statement

This is a problem from Spacetime and Geometry by Carroll,

Just because a manifold is topologically nontrivial doesn't necessarily mean it can't be covered with a single chart. In contrast to the circle ##S^1##, show that the infinite cylinder ##RxS^1## can be covered with just one chart, by explicitly constructing the map.

## Homework Equations

## The Attempt at a Solution

Based on my understanding, a chart is a mapping from an open subset ##U## of a set ##M## to ##R^n## such that the image of ##U## is an open set in ##R^n##.

From my searches, I've came across that I can use the annulus as the open subset because it is homeomorphic to the infinite cylinder, but how do I define/construct the mapping? I think there are a lot of ways?