arXiv subjects

Ryu Ueno

Publications and source records attributed to Ryu Ueno.

5 recordsLinked to original sources

arXiv2026

Doubly Totally-Umbilical Statistical Submanifolds in the Probability Simplex

We give a complete classification of doubly totally-umbilical submanifolds in the probability simplex. The probability simplex is one of the most standard statistical manifolds, and information geometry initiated by S. Amari and H. Nagaoka studies the statistical submanifold theory of the probability simplex. On the other hand, H. Furuhata defined doubly totally-umbilical submanifolds in the geometry of statistical manifolds, inspired by the surface theory of Euclidean space.

math.DG↗

arXiv2025

The Second Variational Formula for Statistical Biharmonic Maps

Recently, the statistical bi-energy functional and its first variational formula were introduced by the author and H. Furuhata. The Maps satisfying the corresponding Euler-Lagrange equation are called statistical biharmonic maps. We present the second variational formula for the statistical bi-energy functional and introduce the notion of stability. If the target statistical manifold is a Hessian manifold, it turns out that the second variational formula can be represented using the Hessian curvature. We provide examples of statistical biharmonic maps into Hessian manifolds that are significant in Hessian and information geometry. We also determine the stability of these statistical biharmonic maps, including the improper affine sphere.

math.DG↗

arXiv2025

Geodesic Connectedness on Statistical Manifolds with Divisible Cubic Forms

The class of statistical manifolds with divisible cubic forms arises from affine differential geometry. We examine the geodesic connectedness of affine connections on this class of statistical manifolds. In information geometry, the geodesic connectedness of the affine connections are often assumed, as in the generalized Pythagorean theorem. In Riemannian geometry, the geodesic connectedness of the Levi-Civita connection follows from its geodesic completeness by the well-known Hopf-Rinow theorem. However, the geodesic connectedness of general affine connections is more challenging to achieve, even for the Levi-Civita connection in pseudo-Riemannian geometry or for affine connections on compact manifolds. By analogy with the Hopf-Rinow theorem in Riemannian geometry, we establish the geodesic connectedness of the affine connections on statistical manifolds with divisible cubic forms from their geodesic completeness. As an application, we establish a Cartan-Hadamard type theorem for statistical manifolds.

math.DG↗

arXiv2024

Statistical Biharmonicity of Identity Maps

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.

math.DG↗

arXiv2023

A Variation Problem for Mappings between Statistical Manifolds

We present statistical biharmonic maps, a new class of mappings between statistical manifolds naturally derived from a variation problem. We give the Euler-Lagrange equation of this problem and prove that improper affine hyperspheres induce examples of such maps.

math.DG↗