$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\TM}{\operatorname{T\M}}$ $\newcommand{\TN}{\operatorname{T\N}}$

This is a cross-post from MSE.

Let $\M,\N$ be Riemannian manifolds, $f:\M \to \N$ smooth. Let $\bigwedge^k df:\Lambda_k(\TM) \to \Lambda_k(f^*{\TN})$ be the induced map on the exterior powers.

Define

$$E_k(f)=\int_{\M} \| \bigwedge^k df\|^2 \text{Vol}_{\M}.$$

**Has this functional been studied for $k>1$?**

(Eular-Lagrange equations, symmetries, stability and regularity of critical points etc)

To be clear, I am looking for a *reference*, not any kind of analysis provided in answer.

(For the interested, I derived the E-L equation for $k=2$ here, the general case is similar).