Total Scalar Curvature and Harmonic Curvature

Abstract

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.

Weakly Einstein Critical Point Equation

“…which appears similar to (8). An argument very similar to that used in the proof of Theorem 1.1 enables us to prove Theorem 1.3.…”

“…Lafontaine showed that the conjecture holds if a solution metric g is conformally flat and the kernel of s ′ * g is nontrivial, or ker s ′ * g = 0 [5]. Recently, Yun, Chang, and Hwang showed that the Besse conjecture is true for Riemannian manifolds with harmonic curvature [8].…”

The critical point equation and contact geometry

“…Recently, Barros and Ribeiro Jr [3] proved that the CPE conjecture is also true for half conformally flat. Another partial proof of the CPE conjecture was presented by Yun, Chang and Hwang [12]. They proved that if (g, λ) is a non-trivial solution of the CPE on an n-dimensional compact Riemannian manifold M and satisfies one of the following conditions (i) Ricci tensor of g is parallel (ii) g has harmonic curvature or (iii) g is conformally flat, then (M, g) is isometric to a standard sphere.…”

Vacuum Static Spaces with Vanishing of Complete Divergence of Weyl Tensor

“…[18]). Let (g, f ) be a non-trivial solution of (1.8) on an n-dimensional compact manifold M , n ≥ 4.…”

“…Note that the function α is defined only on the set M ∖ Crit(f ), where Crit(f ) is the set of all critical points of f . However, since α ≤ z , α can be extended to a C 0 function on the whole manifold M (for more details, see [18]). Therefore, when T = 0, it follows from (5.3) that…”