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 , and formally define the conjugate one-form to be .
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. . Since , from the point of view of complex analysis, the quotient 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 (just as ).
For a given function f, let us write , i.e. , where ∂ denotes the partial derivative. Then . Now d((df)∗) is not always zero, indeed , where .
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 (ω is closed) and (ω∗ is closed). These are called the Cauchy–Riemann equations on . Usually they are expressed in terms of as and .
Notable results
A harmonic differential (one-form) is precisely the real part of an (analytic) complex differential. To prove this one shows that satisfies the Cauchy–Riemann equations exactly when is locally an analytic function of . Of course an analytic function 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 . If ω is a harmonic differential, so is ω∗.
See also De Rham cohomology
References