Solitons on Statistical Manifolds and Their Submanifolds
Abstract
Statistical manifolds, introduced by Amari in information geometry, carry a pair of dual torsion-free connections (∇,∇*) together with a Riemannian metric. In this chapter we survey the theory of \emph{statistical solitons}, that is, vector fields and constants that solve the Ricci-soliton equation simultaneously for ∇ and ∇*, together with the statistical analogue of almost quasi-Yamabe solitons. We discuss two principal settings: (i) statistical warped products R×f N2 and N1×fR for (pseudo) Riemannian manifolds N1 and N2, where the soliton equation constrains the warping function; and (ii) statistical submanifolds of Kenmotsu statistical manifolds, where the position of the Reeb field ξ relative to the submanifold and the presence of concircular or concurrent vector fields determine the Einstein, η-Einstein or quasi-Einstein character of the submanifold. We further review the results of Blaga and Chen on gradient solitons and statistical structures, in particular the connections ∇θ defined by a $1$-form, volume formulas and bounds for the Ricci tensor. We also describe Chen-type inequalities for statistical warped product submanifolds, which give the extrinsic context in which such solitons live. Key steps of the proofs, remarks on the sharpness of hypotheses, worked examples, and open problems are included.
Related
- Paired Statistical Solitons and Torse-Forming Rigidity on Statistical Manifolds
- Gradient Ricci Solitons on PTR-Submanifolds of Complex Space Forms
- Arithmeticity, superrigidity, and totally geodesic submanifolds of infinite-volume hyperbolic manifolds
- Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature
- A Note on Partially Anti-invariant Submanifolds of Kenmotsu Manifolds
BibTeX
@misc{siddiqui2026solitons,
title = {{Solitons on Statistical Manifolds and Their Submanifolds}},
author = {Aliya Naaz Siddiqui and Bang-Yen Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38723},
eprint = {2610.38723},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38723},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Solitons on Statistical Manifolds and Their Submanifolds
AU - Aliya Naaz Siddiqui
AU - Bang-Yen Chen
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38723
UR - https://arxiv.org/abs/2610.38723
AB - Statistical manifolds, introduced by Amari in information geometry, carry a pair of dual torsion-free connections (∇,∇^*) together with a Riemannian metric. In this chapter we survey the theory of \emph{statistical solitons}, that is, vector fields and constants that solve the Ricci-soliton equation simultaneously for ∇ and ∇^*, together with the statistical analogue of almost quasi-Yamabe solitons. We discuss two principal settings: (i) statistical warped products R×_f N_2 and N_1×_fR for (pseudo) Riemannian manifolds N_1 and N_2, where the soliton equation constrains the warping function; and (ii) statistical submanifolds of Kenmotsu statistical manifolds, where the position of the Reeb field ξ relative to the submanifold and the presence of concircular or concurrent vector fields determine the Einstein, η-Einstein or quasi-Einstein character of the submanifold. We further review the results of Blaga and Chen on gradient solitons and statistical structures, in particular the connections ∇^θ defined by a $1$-form, volume formulas and bounds for the Ricci tensor. We also describe Chen-type inequalities for statistical warped product submanifolds, which give the extrinsic context in which such solitons live. Key steps of the proofs, remarks on the sharpness of hypotheses, worked examples, and open problems are included.
ER -