## On the first eigenvalue of the Dirac operator on 6-dimensional manifolds.(English)Zbl 0577.58034

Let $$(M^ n,g)$$ be a closed Riemannian spin-manifold with positive scalar curvature R and let $$R_ 0$$ denote its minimum. If $$\Lambda^{\pm}$$ is the first positive or negative eigenvalue of the Dirac operator on M, then $\sqrt{(n/(n-1))R_ 0}\leq | \Lambda^{\pm}|$ and if equality holds then M must be an Einstein space. The authors give the first example of $$(M^ n,g)$$ with n even different from the sphere realizing the lower bound as an eigenvalue.
Reviewer: K.Wojciechowski

### MSC:

 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C30 Differential geometry of homogeneous manifolds

### Keywords:

Einstein metric; scalar curvature; spin-manifold; Dirac operator
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### References:

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