On a compact n-dimensional manifold, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture was proposed in 1984 by Besse, but has yet to be proved. In this paper, we prove that if the manifold with the critical point metric has harmonic curvature, then it is isometric to a standard sphere.1991 Mathematics Subject Classification. 58E11, 53C25.

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Key words total scalar curvature functional, Einstein metric, second homology MSC (2000) 53C25On 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 ker s * g = 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.

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Reducing the functional equations introduced in Proc. Indian Acad. Sci. (Math. Sci.) 113(2) (2003) 91-98 and in Appl. Math. Lett. 21 (2008) 974-977 to equations in complex variables and quaternions, we find general solutions of the equations.We also obtain the stability of the equations.