Solitons on Statistical Manifolds and Their Submanifolds

Abstract

Statistical manifolds, introduced by Amari in information geometry, carry a pair of dual torsion-free connections $(\nabla,\nabla^{*})$ 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 $\nabla$ and $\nabla^{*}$, together with the statistical analogue of almost quasi-Yamabe solitons. We discuss two principal settings: (i) statistical warped products $\R\times_f N_2$ and $N_1\times_f\R$ 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 $\nabla^θ$ 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.