Oriented straight lines and twistor correspondence

Abstract

The tangent bundle to the $n$--dimensional sphere is the space of oriented lines in ${\mathbb{R}}^{n+1}$. We characterise the smooth sections of $T{S}^n\to S^n$ which correspond to points in ${\mathbb{R}}^{n+1}$ as gradients of eigenfunctions of the Laplacian on $S^n$ with eigenvalue $n$. The special case of $n=6$ and its connection with almost complex geometry is discussed.

8 pages, one figure. Final version, to appear in Geometriae Dedicata