Harmonic unit normal sections of Grassmannians associated with cross products

Abstract

Let $G\left( k,n\right)$ be the Grassmannian of oriented subspaces of dimension k of $\mathbb {R}^{n}$ with its canonical Riemannian metric. We study the energy of maps assigning to each $P\in G\left( k,n\right)$ a unit vector normal to P. They are sections of a sphere bundle $E_{k,n}^{1}$ over $G\left( k,n\right)$. The octonionic double and triple cross products induce in a natural way such sections for $k=2$, $n=7$ and $k=3$, $n=8$, respectively. We prove that they are harmonic maps into $E_{k,n}^{1}$ endowed with the Sasaki metric. This, together with the well-known result that Hopf vector fields on odd dimensional spheres are harmonic maps into their unit tangent bundles, allows us to conclude that all unit normal sections of the Grassmannians associated with cross products are harmonic. In a second instance we analyze the energy of maps assigning an orthogonal complex structure $J\left( P\right)$ on $P^{\bot }$ to each $P\in G\left( 2,8\right)$. We prove that the one induced by the octonionic triple product is a harmonic map into a suitable sphere bundle over $G\left( 2,8\right)$. This generalizes the harmonicity of the canonical almost complex structure of $S^{6}$.