Statistical Biharmonicity of Identity Maps
Ryu Ueno
TL;DR
The paper investigates statistical manifolds (M, g, ∇) whose identity map is biharmonic in the statistical sense, linking biharmonicity to a semi-equiaffine condition expressed by the pair of equations $Δ_g T + \sum_{i=1}^m ext{Ric}^g(e_i,T)e_i=0$ and $\text{div}^g(T)T+\nabla^g_T T=0$ for the Tchebychev vector field $T$. It shows that these equations are equivalent to the vanishing of the statistical bi-tension fields $ au_2( ext{id})$ and $\overline{τ}_2( ext{id})$, and analyzes the consequences in the constant-curvature setting, deriving explicit forms for the statistical structure $(g,∇)$ in complete simply-connected spaces of curvature $c$. In particular, for $M$ modeled on spheres, Euclidean space, or hyperbolic space, the results yield either $T=0$ (with $ abla= abla^g$) or a parallel/tangent-constant $T$ in the Euclidean case, together with algebraic relations tying $λ$, $g(T,T)$, and $g(K,K)$. This provides a concrete classification of semi-equiaffine statistical manifolds under constant curvature and clarifies the role of the Tchebychev field in constraining the statistical structure.
Abstract
The tension field of the identity map from a statistical manifold to a Riemannian statistical manifold, which shares the same Riemannian metric, is the Tchevychev vector field multiplied by negative one. We derive a new class of statistical manifolds that satisfy the semi-equiaffine condition based on the statistical biharmonicity of the identity map. Furthermore, we determine the statistical structures of this class, when the pair of the manifold and the Riemannian metric is a simply connected complete Riemannian manifold of constant curvature.