A quadratic differential on a Riemann surface is a bundle section of the tensor
square of its holomorphic cotangent bundle. In
a local complex coordinate 
, it is written
 |
Under a holomorphic change of coordinate 
,
its coefficient transforms according to
 |
so the expression 
is coordinate independent. The quadratic differential
is called holomorphic or meromorphic
when its local coefficient has the corresponding
property.
Away from its zeros and poles, the quadratic differential determines horizontal and vertical directions by the conditions
that 
be positive or negative real, respectively. Quadratic
differentials occur in the theory of Riemann surfaces,
Teichmüller space, and surface immersions.
The Hopf differential is an important geometric
example.
