An algebraic manifold is another name for a smooth variety. It can be covered by affine open subvarieties, with regular maps
as transition maps on overlaps. These affine open subvarieties need not
be all of affine space 
.
For example, the sphere is an algebraic manifold, with a coordinate chart given by stereographic
projection to 
,
and another coordinate 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. Its complex
structure can be described as the quotient space 
, where 
is a lattice. Local
inverses of the quotient map give analytic coordinate charts. The Weierstrass
elliptic function and its derivative instead give
a parametrization of a smooth
cubic algebraic curve (Milne 2021).
