Harmonic differential

In mathematics, a real differential one-form ω 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 which are the real parts of complex differentials. Let $\scriptstyle \omega \,=\,A\,{\text{d}}x\,+\,B\,{\text{d}}y$, and formally define the conjugate one-form to be $\scriptstyle \omega ^{*}\,=\,A\,{\text{d}}y\,-\,B\,{\text{d}}x$.

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 ƒ, let us write ω = dƒ, i.e. ω = (∂ƒ/∂x) dx + (∂ƒ/∂y) dy where ∂ denotes the partial derivative. Then (dƒ)* = (∂ƒ/∂x) dy − (∂ƒ/∂y) dx. Now d(dƒ)* is not always zero, indeed d(dƒ)* = Δƒ dx dy, where Δƒ = ∂²ƒ/∂x² + ∂²ƒ/∂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 dƒ of solutions ƒ to Laplace's equation Δƒ = 0.

• If ω is a harmonic differential, so is ω*.

See also

• De Rham cohomology