Harmonic differential

Not to be confused with Harmonic form.

In mathematics, a real differential one-form ω on a surface is called a harmonic differential if ω and its conjugate one-form, written as ω^∗, are both closed.

Explanation

Consider the case of real one-forms defined on a two dimensional real manifold. Moreover, consider real one-forms that are the real parts of complex differentials. Let ω = A dx B dy, and formally define the conjugate one-form to be ω^∗ = A dy − B dx.

Motivation

There is a clear connection with complex analysis. Let us write a complex number z in terms of its real and imaginary parts, say x and y respectively, i.e. z = x iy. Since ω iω^∗ = (A − iB)(dx i dy), from the point of view of complex analysis, the quotient (ω iω^∗)/dz tends to a limit as dz tends to 0. In other words, the definition of ω^∗ was chosen for its connection with the concept of a derivative (analyticity). Another connection with the complex unit is that (ω^∗)^∗ = −ω (just as i² = −1).

For a given function f, let us write ω = df, i.e. ω = ⁠∂f/∂x⁠ dx ⁠∂f/∂y⁠ dy, where ∂ denotes the partial derivative. Then (df)^∗ = ⁠∂f/∂x⁠ dy − ⁠∂f/∂y⁠ dx. Now d((df)^∗) is not always zero, indeed d((df)^∗) = Δf dx dy, where Δf = ⁠∂²f/∂x²⁠ ⁠∂²f/∂y²⁠.

Cauchy–Riemann equations

As we have seen above: we call the one-form ω harmonic if both ω and ω^∗ are closed. This means that ⁠∂A/∂y⁠ = ⁠∂B/∂x⁠ (ω is closed) and ⁠∂B/∂y⁠ = −⁠∂A/∂x⁠ (ω^∗ is closed). These are called the Cauchy–Riemann equations on A − iB. Usually they are expressed in terms of u(x, y) iv(x, y) as ⁠∂u/∂x⁠ = ⁠∂v/∂y⁠ and ⁠∂v/∂x⁠ = −⁠∂u/∂y⁠.

Notable results

• A harmonic differential (one-form) is precisely the real part of an (analytic) complex differential. To prove this one shows that u iv satisfies the Cauchy–Riemann equations exactly when u iv is locally an analytic function of x iy. Of course an analytic function w(z) = u iv is the local derivative of something (namely ∫w(z) dz).
• The harmonic differentials ω are (locally) precisely the differentials df of solutions f to Laplace's equation Δf = 0.
• If ω is a harmonic differential, so is ω^∗.

See also

• De Rham cohomology