Einstein manifolds of dimension five with small first eigenvalue of the Dirac operator. (English) Zbl 0633.53069

Friedrich, Th.; Kath, I.

J. Differ. Geom. 29, No. 2, 263-279 (1989).

Let \(M^ n\) be a compact Einstein spin manifold with positive scalar curvature R and denote by \(D: \Gamma\) (S)\(\to \Gamma (S)\) the Dirac operator acting on sections of the spinor bundle. If \(\lambda _ 1\) is the first eigenvalue of this operator we have \(\lambda ^ 2_ 1\geq (1/4)(nR/(n-1))\). Thus, there arises the interesting problem to classify all those Einstein spaces where the lower bound actually is an eigenvalue of the Dirac operator. The corresponding eigenspinor \(\psi\) satisfies the stronger equation \[ \nabla _ X\psi =\mp \sqrt{R/(n(n-1))} X\cdot \psi \] and these spinors are sometimes called Killing spinors. In case \(n\leq 4\) the only possible manifolds are spaces of constant curvature.
The aim of this paper is to study the above mentioned classification problem in the case of 5-dimensional Einstein spaces. It turns out that a Killing spinor defines an Einstein-Sasaki structure on \(M^ 5\) and conversely (in case \(\pi _ 1(M)=0)\). Moreover the paper contains a classification of all regular Einstein-Sasaki 5-manifolds with Killing spinors. The possible spaces are \(S^ 5\), \(S^ 5/Z_ 3\), \(V_{4,2}\), \(V_{4,2}/Z_ 2\) or special \(S^ 1\)-fibre bundles over one of the del- Pezzo surfaces \(P_ k\). There is a one-to-one correspondence between Einstein metrics with Killing spinors on \((S^ 2\times S^ 3)\#...\#(S^ 2\times S^ 3)\) and Einstein-Kähler structures on \(P_ k\).

Reviewer: Th.Friedrich

MSC:

53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
58J50	Spectral problems; spectral geometry; scattering theory on manifolds
58J60	Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)

Keywords:

compact Einstein spin manifold; positive scalar curvature; Dirac operator; first eigenvalue; Einstein spaces; Killing spinors; Einstein- Sasaki 5-manifolds