An algebraic manifold is another name for a smooth algebraic variety. It can be covered by affine open subvarieties, with regular
maps as transition maps on overlaps. These affine charts need not be all of affine
space 
.
For example, the sphere is an algebraic manifold, with a chart given by stereographic projection
to 
,
and another chart at 
,
with the transition function given by 
. In this setting, it is called the
Riemann sphere. The torus
is also an algebraic manifold, in this setting called an elliptic
curve, with charts given by elliptic functions
such as the Weierstrass elliptic function.
