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

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.)
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