Four dimensional static and related critical spaces with harmonic curvature

TL;DR

This paper classifies 4-dimensional Riemannian manifolds with harmonic curvature satisfying a specific differential equation, revealing they are locally isometric to well-known geometric spaces or static metrics, and extends classification results for related geometric structures.

Contribution

It provides a comprehensive classification of 4D harmonic curvature manifolds satisfying a particular PDE, including static, conformally flat, and Ricci-flat spaces, with new corollaries for static and critical metrics.

Findings

Manifolds are locally isometric to five specific types of spaces.

Classification includes static, conformally flat static, and Ricci-flat spaces.

Derived corollaries for static, Miao-Tam critical, and V-static spaces.

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 equation \begin{eqnarray} \label{0002bx} \nabla df = f(Rc -\frac{R}{n-1} g) + x Rc+ y(R) g. \end{eqnarray} where $Rc$ 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; {\rm (i)} $ \mathbb{S}^2(\frac{R}{6}) \times \mathbb{S}^2(\frac{R}{3})$ with $R>0$, {\rm (ii)} $ \mathbb{H}^2(\frac{R}{6}) \times \mathbb{H}^2(\frac{R}{3}) $ with $R<0$, where $\mathbb{S}^2(k) $ and $\mathbb{H}^2(k) $ are the two-dimensional Riemannian manifold with constant sectional curvature $k>0$ and $k<0$, respectively, {\rm (iii)} the static spaces in Example 3 below, {\rm (iv)} conformally flat static spaces described in Kobayashi's \cite{Ko}, and {\rm (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 \cite{Ki}. Some Codazzi-tensor properties of Ricci tensor, which come from the harmonicity of curvature, are effectively used.