In this article we make a classification of four‐dimensional gradient almost Ricci solitons with harmonic Weyl curvature. We prove first that any four‐dimensional (not necessarily complete) gradient almost Ricci soliton $(M,g,f,\lambda )$ with harmonic Weyl curvature has less than four distinct Ricci‐eigenvalues at each point. If it has three distinct Ricci‐eigenvalues at each point, then $(M,g)$ is locally a warped product with 2‐dimensional base in explicit form, and if g is complete in addition, the underlying smooth manifold is ${\mathbb{R}}^2\times M_k^2$ or $\left({\mathbb{R}}^2-\{(0,0)\}\right)\times M_k^2$. Here $M_k^2$ is a smooth surface admitting a complete Riemannian metric with constant curvature k. If $(M,g)$ has less than three distinct Ricci‐eigenvalues at each point, it is either locally conformally flat or locally isometric to the Riemannian product ${\mathbb{R}}^2\times N_{\lambda }^2$, $\lambda \ne 0$, where ${\mathbb{R}}^2$ has the Euclidean metric and $N_{\lambda }^2$ is a 2‐dimensional Riemannian manifold with constant curvature λ. We also make a complete description of four‐dimensional gradient almost Ricci solitons with harmonic curvature.

中文翻译：

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具有谐波Weyl曲率的几乎具有Ricci孤子的四维梯度

在本文中，我们对具有谐波Weyl曲率的近似Ricci孤子的四维梯度进行了分类。我们首先证明任何四次（不一定是完整的）梯度几乎都是里奇孤子$（\mathrm{中号}，G，F，\lambda ）$谐Weyl曲率在每个点上具有少于四个不同的Ricci-本征值。如果每个点都有三个不同的Ricci特征值，则$（\mathrm{中号}，G）$是局部具有明显形式的二维底量的翘曲产品，并且如果g另外完整，则基础光滑流形为${\mathbb{[R}}^{\mathrm{2个}}\times {\mathrm{中号}}_{ķ}^{\mathrm{2个}}$ 或者 $\left({\mathbb{[R}}^{\mathrm{2个}}-\{（0，0）\}\right)\times {\mathrm{中号}}_{ķ}^{\mathrm{2个}}$。这里${\mathrm{中号}}_{ķ}^{\mathrm{2个}}$是一个光滑的表面，它接受具有恒定曲率k的完整黎曼度量。如果$（\mathrm{中号}，G）$ 在每个点上具有少于三个不同的Ricci-本征值，它对于黎曼乘积是局部共形的或局部等距的 ${\mathbb{[R}}^{\mathrm{2个}}\times {ñ}_{\lambda }^{\mathrm{2个}}$， $\lambda \ne 0$， 在哪里 ${\mathbb{[R}}^{\mathrm{2个}}$ 具有欧几里得度量， ${ñ}_{\lambda }^{\mathrm{2个}}$是具有恒定曲率λ的二维黎曼流形。我们还完整地描述了具有谐波曲率的近似Ricci孤子的四维梯度。