广义准统计结构
Generalized quasi-statistical structures
AI总结：
该研究在微分几何框架下引入广义准统计结构概念，刻画了广义切丛上广义复/积结构的∇可积性与准统计流形的等价关系，构造了准统计结构到切/余切丛的延拓并定义了相关几何结构。
摘要：
给定光滑流形M上的非退化(0,2)型张量场h，我们研究M的广义切丛TM⊕T*M上的一类自然广义复结构与广义积结构，并证明：对于M上的仿射联络∇，上述结构是∇可积的当且仅当(M,h,∇)是准统计流形。我们引入广义准统计结构的概念，证明M上的任意准统计结构都能在TM⊕T*M上诱导出广义准统计结构。在此框架下讨论了对偶联络并推导了其部分性质。研究结果结合T*M上的Patterson-Walker度量与Sasaki度量、TM上的水平提升度量与Sasaki度量进行表述；当联络∇平坦时，我们通过广义几何定义了流形上准统计结构到其余切丛与切丛的延拓。此外，还在T*M和TM上定义了Norden结构与Para-Norden结构。
英文摘要：
Given a non-degenerate $(0,2)$-tensor field $h$ on a smooth manifold $M$, we consider a natural generalized complex and a generalized product structure on the generalized tangent bundle $TM\oplus T^*M$ of $M$ and we show that they are $\nabla$-integrable, for $\nabla$ an affine connection on $M$, if and only if $(M,h,\nabla)$ is a quasi-statistical manifold. We introduce the notion of generalized quasi-statistical structure and we prove that any quasi-statistical structure on $M$ induces generalized quasi-statistical structures on $TM\oplus T^*M$. In this context, dual connections are considered and some of their properties are established. The results are described in terms of Patterson-Walker and Sasaki metrics on $T^*M$, horizontal lift and Sasaki metrics on $TM$ and, when the connection $\nabla$ is flat, we define prolongation of quasi-statistical structures on manifolds to their cotangent and tangent bundles via generalized geometry. Moreover, Norden and Para-Norden structures are defined on $T^*M$ and $TM$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
