A note on rigidity of Riemannian manifolds with positive scalar curvature
- PDF / 286,036 Bytes
- 9 Pages / 439.37 x 666.142 pts Page_size
- 1 Downloads / 203 Views
Archiv der Mathematik
A note on rigidity of Riemannian manifolds with positive scalar curvature Guangyue Huang and Qianyu Zeng
Abstract. In this short note, we obtain an integral inequality for closed Riemannian manifolds with positive scalar curvature and give some rigidity characterization of the equality case, which generalizes the recent results of Catino which deal with the conformally flat case, and of Huang and Ma which deal with the harmonic curvature case. Moreover, we obtain an integral pinching condition with non-negative constant σ2 (Aτ ), which can be seen as a complement to Bo and Sheng who considered conformally flat manifolds with constant quotient curvature of σk (Aτ ). Mathematics Subject Classification. Primary 53C20, Secondary 53C21. Keywords. Einstein, Rigidity, Conformally flat.
1. Introduction. Let (M n , g) be an n-dimensional Riemannian manifold with n ≥ 3. It is well-known that for n ≥ 4, the metric g is conformally flat if and only if its Weyl curvature tensor is zero. If n = 3, then it is conformally flat if and only if the Cotton tensor is zero. In the last years, the classifications of conformally flat manifolds under some geometrical or topological assumptions have been paid much attention. For example, in [17], Tani proved that any closed conformally flat manifold with positive Ricci curvature and constant scalar curvature is covered isometrically by Sn with the round metric. For complete conformally flat manifolds with non-negative Ricci curvature, Carron and Herzlich [2] gave the following classifications: they are either flat, or locally isometric to R × Sn−1 with the product metric; or are globally conformally equivalent to Rn or to a spherical space form. For closed conformally flat manifolds satisfying some integral pinching conditions, see [5,7,8,16,18]. On the other hand, for some classifications with point-wise pinching condition on the Ricci curvature, see [4,14,19] and references therein. The research of the authors is supported by NSFC (Nos. 11971153, 11671121).
Arch. Math.
G. Huang and Q. Zeng
Throughout this paper, all the calculations are carried out under the normal coordinates. Denote by R, Rij the scalar curvature and the Ricci curvature ˚ij = Rij − R gij be the trace-less Ricci curvature. With respectively. We let R n the help of the properties of the Codazzi tensor, Catino [3] studied closed conformally flat manifolds with positive constant scalar curvature (in this case, the Ricci curvature is a Codazzi tensor) and satisfying an optimal integral pinching condition. He proved the following Theorem A. Let (M n , g) be a closed conformally flat Riemannian manifold with positive constant scalar curvature. Then ˚ij | |R ˚ij | n−2 n ≤ 0, (1.1) R − n(n − 1)|R M
and equality occurs if and only if (M n , g) is covered isometrically by either Sn with the round metric, S1 × Sn−1 with the product metric, or S1 × Sn−1 with a rotationally symmetric Derdzi´ nski metric. Generalizing the above results of Catino, Huang and Ma [11] studied manifolds with harmonic curvature tenso
Data Loading...
