Four-dimensional static and related critical spaces with harmonic curvature

Abstract

In this article we study any 4-dimensional Riemannian manifold (M, g) with harmonic curvature which admits a smooth nonzero solution f to the following equationwhere Rc is the Ricci tensor of g, x is a constant and y(R) a function of the scalar curvature R. We show that a neighborhood of any point in some open dense subset of M is locally isometric to one of the following five types;and H 2 (k) are the two-dimensional Riemannian manifold with constant sectional curvature k > 0 and k < 0, respectively, (iii) the static spaces in Example 3 below, (iv) conformally flat static spaces described in Kobayashi's [18], and (v) a Ricci flat metric.We then get a number of Corollaries, including the classification of the following four dimensional spaces with harmonic curvature; static spaces, Miao-Tam critical metrics and V -static spaces.The proof is based on the argument from a preceding study of gradient Ricci solitons [17]. Some Codazzi-tensor properties of Ricci tensor, which come from the harmonicity of curvature, are effectively used.where Rc is the Ricci curvature of g, x is a constant and y(R) a function of R. There are several well-known classes of spaces which admit such solutions.

Critical metrics on $4$-manifolds with harmonic anti-self dual Weyl tensor

“…Whence, one sees from (2.3) that div (W ) = 0 in (M 4 , g). Finally, it suffices to apply Theorem 10.2 in [12] to conclude that the manifold M 4 is isometric to a geodesic ball in a simply connected space form R 4 , H 4 or S 4 . So, the proof is finished.…”

“…This result includes, in particular, the half-conformally flat case (i.e., W + = 0). Recently, Kim and Shin [12] showed that a simply connected, compact Miao-Tam critical metric with harmonic Weyl tensor (i.e., div(W ) = 0) and boundary isometric to a standard sphere S 3 must be isometric to a geodesic ball in a simply connected space form R 4 , H 4 or S 4 (see also [2,4] for an alternative proof).…”

Vacuum Static Spaces with Vanishing of Complete Divergence of Weyl Tensor

“…[13]), and in dimension n = 4, it is well known ( [3]) that the Bach tensor is conformally invariant and arises as a gradient of the total Weyl curvature functional, which is given by the integral of the square norm of the Weyl tensor. On the other hand, Kim and Shin proved a local classification of four-dimensional vacuum static spaces with harmonic curvature, or div R = 0 ( [11]). We say that a Riemannian manifold (M, g) has harmonic curvature if div R = 0, or equivalently, that the Ricci tensor r g is a Codazzi tensor.…”

Vacuum Static Spaces with Positive Isotropic Curvature

“…[22]), and in dimension n = 4, it is well known [2] that the Bach tensor is conformally invariant and arises as a gradient of the total Weyl curvature functional, which is given by the integral of the square norm of the Weyl tensor. On the other hand, Kim and Shin [18] proved a local classification of four-dimensional vacuum static spaces with harmonic curvature. We say that a Riemannian manifold (M, g) has harmonic curvature if div R = 0, or equivalently, that the Ricci tensor r g is a Codazzi tensor.…”