AI 中文总结
该研究证明低维广义Ricci孤立子的微分同胚类型，确定三类弦背景的拓扑约束，明确多数弦背景无法与对应经典特殊全纯性流形共存，并刻画BHE三维流形的拓扑。
AI 中文摘要
借鉴Akutagawa等人2007年关于Perelman的研究思路，我们证明紧致广义Ricci孤立子（GRS）具有正Yamabe不变量。作为Wei等人2009年关于Bakry-Émery Ricci曲率分裂定理的推论，我们得到GRS的Cheeger-Gromoll型分裂定理。利用该定理，我们证明低维GRS微分同胚于$S^3/\boldsymbol{\text{Γ}}$或$S^3\times S^1/\boldsymbol{\text{Γ}}$。我们确定弦背景（Bismut-Hermitian-Einstein（BHE）、强挠率$G_2$、强挠率$\text{Spin}(7)$流形）的多种拓扑约束，并证明在多数情况下，它们无法与对应的经典特殊全纯性对应物存在于同一流形上。最后，我们借助Kollar 2006年关于复轨形上Seifert纤维化5维流形的刻画的部分扩展，确定自然约束下BHE三维流形的拓扑。
英文摘要
Adapting ideas of \cite{akutagawa2007perelman}, we show that compact generalized Ricci solitons (GRS) have positive Yamabe invariant. We observe a Cheeger-Gromoll-type splitting theorem for GRS as a corollary of the splitting theorem for Bakry-Émery Ricci curvature in \cite{wei2009comparison}. Using this we show that low dimensional GRS are diffeomorphic to $S^3 / Γ$ or $S^3 \times S^1 / Γ$. We determine various topological constraints on string backgrounds (Bismut-Hermitian-Einstein (BHE), strong torsion $G_2$, strong torsion $\mathrm{Spin}(7)$-manifolds) and show in most cases that they cannot exist on the same manifolds as their classical special holonomy counterparts. Finally we determine the topology of BHE threefolds under natural constraints, relying on an extension of parts of Kollar's characterization of Seifert fibered $5$-manifolds over complex orbifolds \cite{kollar2006circle}.
