The Hopf differential of an oriented conformal immersion 
is the quadratic
differential given in a local complex coordinate 
by
 |
where 
is the ambient Levi-Civita connection,
is the unit
normal vector, and the brackets denote the ambient inner
product. Equivalently, it is the 
-part of the second
fundamental form. The transformation law for a quadratic
differential makes this definition independent of the chosen conformal coordinate.
The zeros of the Hopf differential are the umbilic points of the immersion. For a surface in a 3-dimensional manifold of constant sectional curvature, the Codazzi equations imply that the Hopf differential is holomorphic exactly when the mean curvature is constant. In particular, the Hopf differential of a minimal surface is holomorphic.
