An algebraic manifold is another name for a smooth algebraic variety. It can be covered by coordinate charts
so that the transition functions are given
by rational functions. Technically speaking,
the coordinate charts should be to 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.
