Rigidity of the critical point equation

초록

On a compact n-dimensional manifold M, it was shown that a critical point metric g of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satisfies the critical point equation ([5], p. 3222). In 1987 Besse proposed a conjecture in his book [1], p. 128, that a solution of the critical point equation is Einstein (Conjecture A, hereafter). Since then, number of mathematicians have contributed for the proof of Conjecture A and obtained many geometric consequences as its partial proofs. However, none has given its complete proof yet. The purpose of the present paper is to prove Theorem 1, stating that a compact 3-dimensional manifold M is isometric to the round 3-sphere S(3) if s(g)'* not equal 0 and its second homology vanishes. Note that this theorem implies that M is Einstein and hence that Conjecture A holds on a 3-dimensional compact manifold under certain topological conditions. (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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