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

Abstract

We study statistical manifolds (M,g,∇,∇^*) endowed with a nonzero (1,1)-tensor field Θ satisfying the gauge equation ∇_X(ΘY)=Θ(∇_X^*Y). We first characterize this condition in terms of the statistical difference tensor K=∇−∇^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 ImΘ determine smooth integrable distributions. If, in addition, TM=kerΘoverset⊥⊕ImΘ, 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.

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BibTeX RIS

BibTeX

@misc{milijevic2026gaugecompatible,
  title = {{Gauge-compatible tensors on statistical manifolds: splitting and submanifold geometry}},
  author = {Mirjana Milijevic and Luis P. Yapu},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.31145},
  eprint = {2608.31145},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.31145},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Gauge-compatible tensors on statistical manifolds: splitting and submanifold geometry
AU  - Mirjana Milijevic
AU  - Luis P. Yapu
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.31145
UR  - https://arxiv.org/abs/2608.31145
AB  - We study statistical manifolds (M,g,∇,∇^*) endowed with a nonzero (1,1)-tensor field Θ satisfying the gauge equation ∇_X(ΘY)=Θ(∇_X^*Y). We first characterize this condition in terms of the statistical difference tensor K=∇−∇^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 ImΘ determine smooth integrable distributions. If, in addition, TM=kerΘoverset⊥⊕ImΘ, 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.
ER  -

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