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.
We study the Yamabe invariant of manifolds obtained as connected sums along submanifolds of codimension greater than 2. In particular: for a compact connected manifold M with no metric of positive scalar curvature, we prove that the Yamabe invariant of any manifold obtained by performing surgery on spheres of codimension greater than 2 on M is not smaller than the invariant of M .
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.
Abstract. We show that given n and D, v > 0, there exists a positive number = (n, D, v) > 0 such that if a closed n-manifold M satisfies Ric(
IntrooductionIn this note, we give some properties of the fundamental groups of Riemannian manifolds of almost nonnegative curvature. In Recall that by a theorem of Gromov a finitely generated group is of polynomial growth if and only if it is almost nilpotent, i.e., it contains a nilpotent subgroup of finite index ([9]). So, together with this, π 1 (M ) is almost nilpotent in Wei's theorem.On the other hand, note that for any nilmanifold N n which is not a torus, π 1 (N ) has polynomial growth of order > n ([10], [13]). Thus we might expect a little stronger conclusion of Wei's theorem.To extend the well-known results on manifolds of nonnegative Ricci curvature to manifolds of almost nonnegative Ricci curvature, a major problem is the splitting conjecture by Fukaya and Yamaguchi ([7], [15]). Recently, Cheeger and Colding ([3]) have announced that they have solved this conjecture; see also [5] for other results on manifolds with almost nonnegative Ricci curvature.
Theorem 1 (Splitting Theorem). Let (X, p) be the pointed Hausdorff limit of a sequenceThen the splitting theorem holds for X, i.e., if X contains a line, then X splits R×X isometrically.Using this theorem together with other results, we can prove the following theorem.
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