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Kähler Manifolds with Small Eigenvalues of the Dirac Operator and a Conjecture of Lichnerowicz

Domenii publicaţii > Matematica + Tipuri publicaţii > Articol în revistã ştiinţificã

Autori: A. Moroianu

Editorial: Ann. Inst. Fourier, 49, p.1637-1659, 1999.


We describe all compact spin K{„a}hler manifolds
$(M^{2m},g,J)$ of even complex dimension and positive scalar curvature
with least possible first eigenvalue of the Dirac operator. More
precisely, we prove a result conjectured by Lichnerowicz, asserting
that if there exists an eigenvalue $l$ of the Dirac operator on $M$
such that $lambda^2=frac{m}{4(m-1)}inf_M S$, (where $S$ the scalar
curvature of $M$), then the universal cover of $M$ is isometric to a
Riemannian product $Nx RM^2$, where $N$ is a limiting manifold of
odd complex dimension $m-1$. We then prove that the above
equality holds if and only if $M$ is the suspension over a flat
parallelogram of two commuting isometries of $N$ preserving a
K{„a}hlerian Killing spinor.

Cuvinte cheie: Kirchberg's Inequality, Kaehlerian Killing Spinors