maximal smooth atlas
An atlas is a collection of charts that cover the entire topological manifold.
A smooth atlas includes a condition that the overlap of charts be compatible with calculus how we know it in
Definition
A maximal smooth atlas for the manifold
, the map is smooth. Note we only care about the intersections. If something lives outside the intersections, i.e. it is only in the “middle” of a chart, then the calculus already works well because it is homeomorphic to inside this openset. In other words, we want calculus to work nice as we move between charts. - Maximality - nothing is left out.
Given a homeomorphism
between open sets in and , such that