An abstract manifold is a manifold in the context of an abstract space with no particular embedding, or representation in mind. It is a topological space with an atlas of coordinate charts.
For example, the sphere 
can be considered a submanifold
of 
or a quotient space 
. But as an abstract manifold, it is just a manifold,
which can be covered by two coordinate charts 
and 
, with the single transition
function,
 |
defined by
 |
where 
.
It can also be thought of as two disks glued together at their boundary.
