统计流形上的规范兼容张量：分裂与子流形几何
Gauge-compatible tensors on statistical manifolds: splitting and submanifold geometry
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中文总结 AI 辅助
该研究针对带规范兼容张量的统计流形，刻画其等价条件，建立局部乘积分解，推导子流形相关方程并给出实例，拓展了统计流形的几何分析。
中文摘要 AI 辅助
我们研究带有非零(1,1)型张量场Θ的统计流形(M,g,∇,∇*)，该张量满足规范方程∇_X(ΘY)=Θ(∇_X^*Y)。我们首先利用统计差张量K=∇−∇^g刻画该条件；当Θ关于Levi-Civita联络平行时，规范方程等价于K_XΘ=−ΘK_X。我们进一步证明Θ可耦合对偶联络的平行迁移，因此其在每个连通分支上秩为常数，kerΘ与ImΘ确定光滑可积分布。若TM=kerΘ⊥⊕ImΘ，我们建立统计结构的局部乘积分解。随后研究带有Θ-不变和Θ-反不变分布的子流形，结合对偶统计联络的第二基本形式与形状算子，推导环境规范方程的切向与法向分量，得到曲率耦合结果，并给出非全测地例子说明子流形恒等式。
英文摘要
We study statistical manifolds $(M,g,\nabla,\nabla^{*})$ endowed with a nonzero $(1,1)$-tensor field $Θ$ satisfying the gauge equation \[ \nabla_X(ΘY)=Θ(\nabla_X^{*}Y). \] We first characterize this condition in terms of the statistical difference tensor \[ K=\nabla-\nabla^{g}. \] When $Θ$ is parallel with respect to the Levi-Civita connection, the gauge equation is equivalent to \[ K_XΘ=-ΘK_X. \] We further show that $Θ$ intertwines the parallel transports of the dual connections. Consequently, its rank is constant on every connected component, and $\kerΘ$ and $\operatorname{Im}Θ$ determine smooth integrable distributions. If, in addition, \[ TM=\kerΘ\overset{\perp}{\oplus}\operatorname{Im}Θ, \] we establish a local product decomposition of the statistical structure. We then study submanifolds carrying $Θ$-invariant and $Θ$-anti-invariant distributions and derive the tangential and normal components of the ambient gauge equation in terms of the second fundamental forms and shape operators of the dual statistical connections. We also obtain curvature-intertwining consequences and present a non-totally-geodesic example illustrating the submanifold identities.
