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


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