On eigenmaps between spheres. (English) Zbl 1033.58014
A \(\lambda_{2}\)-eigenmap \(f: {\mathbb S}^{m-1} \to {\mathbb S}^{n-1}\) is the restriction to the unit sphere of harmonic homogeneous polynomials of degree two. Equivalently, its coordinates are eigenfunctions of the Laplacian on \({\mathbb S}^{m-1}\), corresponding to the second (non-zero) eigenvalue. Calabi proved that a full (i.e. not contained in a proper great sphere of the target) \(\lambda_{2}\)-eigenmap from \({\mathbb S}^{2}\) is equivalent to the Veronese map. While this rigidity fails for higher dimensions, the article under review shows \(\lambda_{2}\)-eigenmaps from \({\mathbb S}^{4}\) into itself to be, up to orthogonal transformations, the gradient map of the cubic isoparametric polynomial of Cartan.
The existence of non-constant \(\lambda_{2}\)-eigenmaps from \({\mathbb S}^{2n-k}\) to \({\mathbb S}^{n}\), for \(k=1,\dots,5\), is investigated and shown to be limited: for \(k=1\), \(\lambda_{2}\)-eigenmaps from \({\mathbb S}^{2n-1}\) to \({\mathbb S}^{n}\) exist only if \(n=1,2,4\) or \(8\) and are essentially the Hopf maps; from \({\mathbb S}^{2n-2}\) to \({\mathbb S}^{n}\) they do not exist; from \({\mathbb S}^{2n-3}\) to \({\mathbb S}^{n}\), \(\lambda_{2}\)-eigenmaps can only exist if \(n=4\) or \(8\) and must be the Hopf constructions on an orthogonal multiplication; from \({\mathbb S}^{2n-4}\) to \({\mathbb S}^{n}\), \(n\) must be 4 or 8; and from \({\mathbb S}^{2n-5}\) to \({\mathbb S}^{n}\) (\(n\geq 6\)), \(n\) must be 6, 8, 9 or 10.
This classification relies on a recent result of Tang, which states that a \(\lambda_{2}\)-eigenmap from \({\mathbb S}^{m-1}\) to \({\mathbb S}^{n-1}\) must be homotopic to a Hopf construction and have great spheres of dimension less than \(m/2 -1\) as fibres. The arguments invoked are topological in nature, as they combine Adams’ result on the number of orthogonal vector fields on a sphere, the Hurwirtz-Radon classification of orthogonal multiplications, the K-theory of \({\mathbb R}P^{n}\) and the Gysin sequence of certain fibrations.
Finally, new examples are constructed, showing that there exist full \(\lambda_{2}\)-eigenmaps from \({\mathbb S}^{4}\) to \({\mathbb S}^{n}\) for all values of \(n\), except possibly \(n=5\) or \(6\), and from \({\mathbb S}^{7}\) to \({\mathbb S}^{n}\), \(n=6\) remaining unsettled.
The existence of non-constant \(\lambda_{2}\)-eigenmaps from \({\mathbb S}^{2n-k}\) to \({\mathbb S}^{n}\), for \(k=1,\dots,5\), is investigated and shown to be limited: for \(k=1\), \(\lambda_{2}\)-eigenmaps from \({\mathbb S}^{2n-1}\) to \({\mathbb S}^{n}\) exist only if \(n=1,2,4\) or \(8\) and are essentially the Hopf maps; from \({\mathbb S}^{2n-2}\) to \({\mathbb S}^{n}\) they do not exist; from \({\mathbb S}^{2n-3}\) to \({\mathbb S}^{n}\), \(\lambda_{2}\)-eigenmaps can only exist if \(n=4\) or \(8\) and must be the Hopf constructions on an orthogonal multiplication; from \({\mathbb S}^{2n-4}\) to \({\mathbb S}^{n}\), \(n\) must be 4 or 8; and from \({\mathbb S}^{2n-5}\) to \({\mathbb S}^{n}\) (\(n\geq 6\)), \(n\) must be 6, 8, 9 or 10.
This classification relies on a recent result of Tang, which states that a \(\lambda_{2}\)-eigenmap from \({\mathbb S}^{m-1}\) to \({\mathbb S}^{n-1}\) must be homotopic to a Hopf construction and have great spheres of dimension less than \(m/2 -1\) as fibres. The arguments invoked are topological in nature, as they combine Adams’ result on the number of orthogonal vector fields on a sphere, the Hurwirtz-Radon classification of orthogonal multiplications, the K-theory of \({\mathbb R}P^{n}\) and the Gysin sequence of certain fibrations.
Finally, new examples are constructed, showing that there exist full \(\lambda_{2}\)-eigenmaps from \({\mathbb S}^{4}\) to \({\mathbb S}^{n}\) for all values of \(n\), except possibly \(n=5\) or \(6\), and from \({\mathbb S}^{7}\) to \({\mathbb S}^{n}\), \(n=6\) remaining unsettled.
Reviewer: Eric Loubeau (Brest)
