Critical points of the total scalar curvature functional on the space of metrics of constant scalar curvature.(English)Zbl 0972.58009

Summary: It is well known that critical points of the total scalar curvature functional $${\mathcal S}$$ on the space of all smooth Riemannian structures of volume 1 on a compact manifold $$M$$ are exactly the Einstein metrics. When the domain of $${\mathcal S}$$ is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is also Einstein or isometric to a standard sphere.
In this paper we prove that $$n$$-dimensional critical points have vanishing $$n-1$$ homology under a lower Ricci curvature bound for dimension less than 8.

MSC:

 58E11 Critical metrics 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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