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 $\scriptstyle z$ in terms of its real and imaginary parts, say $\scriptstyle x$ and $\scriptstyle y$ respectively, i.e. $\scriptstyle z\,=\,x\,+\,iy$. Since $\scriptstyle \omega \,+\,i\omega ^{*}\,=\,(A\,-\,iB)({\text{d}}x\,+\,i{\text{d}}y)$, from the point of view of complex analysis, the quotient $\scriptstyle (\omega \,+\,i\omega ^{*})/{\text{d}}z$ tends to a limit as $\scriptstyle {\text{d}}x$ tends to 0. In other words, the definition of $\scriptstyle \omega ^{*}$ was chosen for its connection with the concept of a derivative (analyticity). Another connection with the complex unit is that $\scriptstyle (\omega ^{*})^{*}\,=\,-\omega$ (just as $\scriptstyle i^{2}\,=\,-1$).

For a given function $\scriptstyle f$, let us write $\scriptstyle \omega \,=\,{\text{d}}\!f$, i.e. $\scriptstyle \omega \,=\,(\partial \!f/\partial x)\,{\text{d}}x\,+\,(\partial \!f/\partial y)\,{\text{d}}y$ where $\scriptstyle \partial$ denotes the partial derivative. Then $\scriptstyle ({\text{d}}\!f)^{*}\,=\,(\partial \!f/\partial x)\,{\text{d}}y\,-\,(\partial \!f/\partial y)\,{\text{d}}x.$ Now $\scriptstyle {\text{d}}({\text{d}}\!f)^{*}$ is not always zero, indeed $\scriptstyle {\text{d}}({\text{d}}\!f)^{*}\,=\,\Delta \,{\text{d}}x\,{\text{d}}y,$ where $\scriptstyle \Delta f\,=\,\partial ^{2}\!f/\partial x^{2}\,+\,\partial ^{2}\!f/\partial y^{2}$

Cauchy–Riemann equations

As we have seen above: we call the one-form $\scriptstyle \omega$ harmonic if both $\scriptstyle \omega$ and $\scriptstyle \omega ^{*}$ are closed. This means that $\scriptstyle \partial \!A/\partial y\,=\,\partial B/\partial x$ ($\scriptstyle \omega$ is closed) and $\scriptstyle \partial B/\partial y\,=\,-\partial \!A/\partial x$ ($\scriptstyle \omega ^{*}$ is closed). These are called the Cauchy–Riemann equations on $\scriptstyle A\,-\,iB.$ Usually they are expressed in terms of $\scriptstyle u(x,y)\,+\,iv(x,y)$ as $\scriptstyle \partial u/\partial x\,=\,\partial v/\partial y$ and $\scriptstyle \partial u/\partial y\,=\,-\partial v/\partial x.$

Notable results

• A harmonic differential (one-form) is precisely the real part of an (analytic) complex differential. To prove this one shows that $\scriptstyle u\,+\,iv$ satisfies the Cauchy–Riemann equations exactly when $\scriptstyle u\,+\,iv$ is locally an analytic function of $\scriptstyle x\,+\,iy$. Of course an analytic function $\scriptstyle w(z)\,=\,u\,+\,iv$ is the local derivative of something (namely $\scriptstyle \int \!w(z)\,{\text{d}}z$)

• The harmonic differentials $\scriptstyle \omega$ are (locally) precisely the differentials $\scriptstyle {\text{d}}\!f$ of solutions $\scriptstyle f$ to Laplace's equation $\scriptstyle \Delta f\,=\,0.$

• If $\scriptstyle \omega$ is a harmonic differential, so is $\scriptstyle \omega ^{*}.$

See also

• De Rham cohomology