Gauge-compatible tensors on statistical manifolds: splitting and submanifold geometry

Abstract

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.