$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\TM}{\operatorname{T\M}}$ $\newcommand{\TN}{\operatorname{T\N}}$ $\newcommand{\sAverage}[1]{\langle#1\rangle} $ $\newcommand{\IP}[2]{\sAverage{#1,#2}}$ $\newcommand{\Cof}{\text{Cof}}$

Lately I derived the following equation (details are at the end):

Theorem:Let $(\M,g),(\N,h)$ be smooth $d$-dimensional oriented Riemannian manifolds,Let $\,f:\M \to \N\,$ be a smooth conformal map. Then

$$ \det df \cdot \delta(df) + (\frac{2}{d}-1) \tr_{g}\big(df \otimes d \det(df) \big) =0. \tag{1} $$

($\delta$ is the adjoint of the pullback connection on $f^*{\TN}$. The equation is an equality of sections of $f^*{\TN}$.)

*Equation $(1)$ has a few immediate (well-known) corollaries:*

- In dimension $d=2$, conformal mappings are harmonic.
- Homotheties are harmonic (in every dimension).
- In dimension $d > 2$, conformal harmonic maps are homotheties (assuming $\M$ is connected): Let $e_i$ be an orthonormal frame for $\TM$. Then $$ 0=\tr_{g} \big( df \otimes d \det(df) \big)= \sum_{i=1}^d d \det(df)(e_i) \cdot df(e_i) .$$ Since $f$ is an immersion, $df(e_i)$ is a basis of $f^*\TN$.

(Summary: A conformal map is harmonic if and only if $d=2$ or it is a homothety).

**Questions:**

**1.** Is equation $(1)$ known? (Perhaps in a Euclidean setting? Or in coordinates?)

**2.** Is equation $(1)$ an Euler-Lagrange's equation of some functional?

*Sketch of derivation:*

My colleagues and I showed the following:

Theorem(2):Let $(\M,g),(\N,h)$ be smooth $d$-dimensional oriented Riemannian manifolds,Let $\,f:\M \to \N\,$ be an

arbitrarysmooth map. Then$$ \delta (\Cof df)=0. \tag{2}$$ where $\Cof df:= (-1)^{d-1} \star_{f^*TN}^{d-1} (\wedge^{d-1} df) \star_{TM}^1.$ (It's a section of $T^*\M \otimes f^*{\TN}$).

(For a proof see Proposition 3.4 here).

Now, if $f$ is conformal, then $$ \Cof df=(\det df)^{1-\frac{2}{d}} df. \tag{3}$$

Plug $(3)$ into $(2)$ (the rest is Leibniz rule + simplification).