. Aim of this note is to present short and natural proofs for the
lower bounds of the spectrum of the Dirac operator on compact Riemannian
and Kahler manifolds with positive scalar curvature.
Keywords: Dirac operator, eigenvalue, lower bound
1991 MSC: 53A50, 53C25, 58G25, 58G30
For Riemannian manifolds a lower bound of the spectrum of the Dirac operator
was first given by Th. Friedrich (c.f.: [Fri80]). The corresponding result in the
Kahler case is due to K.-D. Kirchberg (c.f.: [Kir86] and [Kir90] ). A modified
proof was given by O. Hijazi (c.f. [Hij94]). The idea of the proof in this note is
to estimate the norm of the covariant derivative of an eigenspinor using the direct
sum decomposition of the tensor product of spinor bundle and cotangent bundle.
To achieve this we use the explicit formula for the embedding of the spinor bundle
S into the tensor product
TM (with modifications for Kahler manifolds).
2. Riemannian manifolds
; g) be a compact R...
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