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An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold admitting parallel one-form. (English) Zbl 0934.58026

Let \(M\) be a compact Riemannian spin manifold of dimension \(n\) which has positive scalar curvature \(s\). Let \(S:=\inf_M s\).
Th. Friedrich [Math. Nachr. 97, 117-146 (1980; Zbl 0462.53027)] proved that any eigenvalue \(\lambda\) of the Dirac operator satisfies the inequality \(\lambda^2\geq{nS\over 4(n-1)}\). If \(M\) admits a parallel one form, the authors improve this inequality in Theorem 1.1 by showing \(\lambda^2\geq{(n-1)S\over 4(n-2)}\). The authors study the limiting case in which equality holds showing that if \(\widetilde M\) is the universal cover of \(M\), then \(\widetilde M\) is isometric to a Riemannian product \(\widetilde F\times R\) where \(\widetilde F\) is a simply connected compact Einstein manifold of dimension \((n-1)\) admitting a real Killing spinor. These manifolds have been classified by C. Bär [Commun. Math. Phys. 154, No. 3, 509-521 (1993; Zbl 0778.53037)].
Reviewer: P.Gilkey (Eugene)

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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