Hessian manifolds of positive constant Hessian sectional curvature
Abstract
Hessian structure on a manifold is a geometric structure consisting of a pair (∇,g) of an affine connection ∇ and a Riemannian metric g satisfying certain compatibility conditions. It is also known as a dually flat structure in information geometry. Hessian geometry is considered to be a real analogue of Kähler geometry; in particular, Hessian sectional curvature corresponds to holomorphic sectional curvature in Kähler geometry. The standard models of Hessian manifolds with constant Hessian sectional curvature were classified by Furuhata and Kurose (2013) in the nonpositive curvature case, whereas the positive curvature case is open. In this paper, we classify the standard models in the positive curvature case. For this end, we introduce Hessian structure on quotient manifolds of Kähler manifolds equipped with suitable Hamiltonian actions of abelian Lie groups and realizing the standard models as such quotient manifolds.
Related
- Equivalence on exponential families as Hessian manifolds and classification of exponential families of order 1 with constant Hessian sectional curvature
- Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature
- Hermitian manifolds with constant holomorphic sectional curvature or real bisectional curvature
- Einstein four-manifolds of positive sectional curvature and ADM mass
- Four-dimensional pseudo-umbilical biharmonic submanifolds in six-dimensional Riemannian manifolds of constant sectional curvature
BibTeX
@misc{sakamoto2026hessian,
title = {{Hessian manifolds of positive constant Hessian sectional curvature}},
author = {Hakobi Sakamoto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01737},
eprint = {2610.01737},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01737},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hessian manifolds of positive constant Hessian sectional curvature
AU - Hakobi Sakamoto
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01737
UR - https://arxiv.org/abs/2610.01737
AB - Hessian structure on a manifold is a geometric structure consisting of a pair (∇,g) of an affine connection ∇ and a Riemannian metric g satisfying certain compatibility conditions. It is also known as a dually flat structure in information geometry. Hessian geometry is considered to be a real analogue of Kähler geometry; in particular, Hessian sectional curvature corresponds to holomorphic sectional curvature in Kähler geometry. The standard models of Hessian manifolds with constant Hessian sectional curvature were classified by Furuhata and Kurose (2013) in the nonpositive curvature case, whereas the positive curvature case is open. In this paper, we classify the standard models in the positive curvature case. For this end, we introduce Hessian structure on quotient manifolds of Kähler manifolds equipped with suitable Hamiltonian actions of abelian Lie groups and realizing the standard models as such quotient manifolds.
ER -