For example, in Riemann surface theory, the deformation theory of complex structures is studied classically by means of quadratic differentials ( namely sections of " L " ( " K " 2 ) ).
12.
If a point of Teichm�ller space is represented by a Riemann surface " R ", then the cotangent space at that point can be identified with the space of quadratic differentials at " R ".
13.
According to the definition of a hypercycle, it is a nonlinear, dynamic system, and, in the simplest case, it can be assumed that it grows at a rate determined by a system of quadratic differential equations.
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Since the Riemann surface has a natural hyperbolic metric, at least if it has negative Euler characteristic, one can define a Hermitian inner product on the space of quadratic differentials by integrating over the Riemann surface.
15.
As in the case of translation surfaces there is an analytic interpretation : a half-translation surface can be interpreted as a pair ( X, \ phi ) where X is a Riemann surface and \ phi a quadratic differential on X.
16.
To pass from the geometric picture to the analytic picture one simply takes the quadratic differential defined locally by ( dz ) ^ 2 ( which is invariant under half-translations ), and for the other direction one takes the Riemannian metric induced by \ phi, which is smooth and flat outside of the zeros of \ phi.
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