A biholomorphic map is a holomorphic map 
between complex
manifolds that is a bijection and whose inverse
function 
is also a holomorphic map. Two complex
manifolds admitting such a map are biholomorphic. Biholomorphic maps preserve
their complex structures (Huybrechts 2005).
For example, every invertible complex linear transformation of 
is biholomorphic. In complex dimension one,
a biholomorphic map is a one-to-one conformal mapping
onto its image, establishing conformal equivalence
of the Riemann surfaces. The definition also applies
in higher complex dimensions, where it requires
holomorphicity of both the map and its inverse, rather than preservation of Euclidean
angles.
