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## Harmonic unit normal sections of the Grassmannian

### Ruth Paola Moas

#### Universidad Nacional de Córdoba - Universidad Nacional de Río Cuarto, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakbad21302730fbaf95648ea393ac71272').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addybad21302730fbaf95648ea393ac71272 = 'p&#97;&#111;m&#111;&#97;s' + '&#64;'; addybad21302730fbaf95648ea393ac71272 = addybad21302730fbaf95648ea393ac71272 + '&#117;nc' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r'; var addy_textbad21302730fbaf95648ea393ac71272 = 'p&#97;&#111;m&#111;&#97;s' + '&#64;' + '&#117;nc' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r';document.getElementById('cloakbad21302730fbaf95648ea393ac71272').innerHTML += '<a ' + path + '\'' + prefix + ':' + addybad21302730fbaf95648ea393ac71272 + '\'>'+addy_textbad21302730fbaf95648ea393ac71272+'<\/a>';

Let $G\left( k,n\right)$ be the Grassmannian of oriented subspaces of $% \mathbb{R}^{n}$ of dimension $k$ with its canonical symmetric Riemannian metric. We study the energy of maps assigning a unit vector normal to $P$ to each $P\in G\left( k,n\right)$. 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. We also show that these sphere bundles do not have parallel sections, which trivially would have had minimum energy.

In a second intance we analyze the energy of maps assigning an orthogonal complex structure $J\left( P\right)$ on $P^{\bot }$ to each $P\in G\left( 2,8\right)$. They are sections of the unit sphere bundle over $G\left( 2,8\right)$ whose fiber at each $P$ consists essentially of the skew-symmetric transformations on $P^{\bot }$. We prove that the section naturally induced by the octonionic triple product is a harmonic map. We comment on the relationship with the harmonicity of the canonical almost complex structure of $S^{6}$.

Joint work with Francisco Ferraris (Universidad Nacional de Córdoba, Argentina) and Marcos Salvai (Conicet - Universidad Nacional de Córdoba, Argentina).