AI 中文总结
本文受埃尔米特几何中Bismut的启发，研究带挠率的Sasaki流形，引入∇-爱因斯坦流形概念，推导带挠率Sasaki结构的流并证明其与广义里奇流规范等价。
AI 中文摘要
受埃尔米特几何中Bismut类比的启发，我们研究带挠率的Sasaki流形。特别地，我们刻画了类余凯勒的平坦带挠率Sasaki流形，并引入∇-爱因斯坦流形的概念，将其作为Bismut埃尔米特-爱因斯坦条件的奇数维类比。我们给出非紧例子，并研究了5维和7维的紧∇-爱因斯坦流形。我们还建立了近接触度量结构几何流的一般框架，特别地，推导了带挠率Sasaki结构的流，该流保持挠率闭合的强条件；此外，我们证明该流与广义里奇流是规范等价的。
英文摘要
Motivated by the analogy with the Bismut connection in Hermitian geometry, we study Sasaki with torsion manifolds. In particular, we characterize co-Kähler-like and flat Sasaki with torsion manifolds, and we introduce the notion of a $\nabla$-Einstein manifold as the odd-dimensional analogue of the Bismut Hermite-Einstein condition. We provide non-compact examples and we study compact $\nabla$-Einstein manifold in dimension $5$ and $7$. We also develop a general framework for geometric flows of almost contact metric structures. In particular, we derive a flow for Sasaki with torsion structures that preserves the strong condition, that is, the closure of the torsion. Furthermore we prove that such flow is gauge-equivalent to the generalized Ricci flow and it is gauge-equivalent to the pluriclosed flow, after performing a trivial product with $S^1$.
Comments45 pages. Added new results on the flow for Sasaki with torsion structures
