Kähler manifolds with small eigenvalues of the Dirac operator and a conjecture of Lichnerowicz

Andrei Moroianu

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The author studies the holonomy group of a simply connected indecomposable and reducible Lorentzian spin manifold under the condition that they admit parallel spinors. He shows that there are only two possible situations: either the manifold is a so-called Brinkmann wave or it has Abelian holonomy and is a pp-manifold – a generalization of a plane-wave. The author gives also sufficient conditions for a Brinkmann wave to have as holonomy the semidirect product of holonomy group of a Riemannian...

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A spectral estimate for the Dirac operator on Riemannian flows

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We give a new upper bound for the smallest eigenvalues of the Dirac operator on a Riemannian flow carrying transversal Killing spinors. We derive an estimate on both Sasakian and 3-dimensional manifolds, and partially classify those satisfying the limiting case. Finally, we compare our estimate with a lower bound in terms of a natural tensor depending on the eigenspinor.