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A remark on the first eigenvalue of the Dirac operator on 4-dimensional manifolds

✍ Scribed by Thomas Friedrich

Publisher: John Wiley and Sons
Year: 1981
Tongue: English
Weight: 159 KB
Volume: 102
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.19811020106

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✦ Synopsis

By THOMAS FRIEDRICH of Berlin

(Eingegangen am 9.9. 1980) Let M* he a cony'act RIEMANNian spin inanifold with positive scalar curvature H and let R, denote its minimum. Consider the DIRAC operator D : r ( S ) + r ( S ) acting on sections of the associated spinor bundle S. If I.* is the first positive or negative eigenvalue of t,his operator, then the inequality holds. Furthermore, if R, or -A I / & R, is an eigenvalue of the 2 n -1 DIRAC operator, then M* must he an EINSTEIN space (see [I]). In the case of dimension three we proved that on S3/r the lower bound is an eigenvalue of the operator D if and only if S 3 / r is homogeneous. On the other hand, if n = 5 , we have an EINSTEIN metric of poaitive scalar curvature on the STIEFEL manifold V4,* = S0(4)/80(2) such that

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