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 
, where 
and 
. On the overlap, the
transition function
 |
is defined by
 |
where 
. It can also be thought of as two disks
glued together at their boundary.
