Year: 2025
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Ueno, Ryu
Institution: Hokkaido University
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Award: G7427681983
OpenAlex ID: W4416038301
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