scholarly journals Statistical Manifolds with almost Quaternionic Structures and Quaternionic Kähler-like Statistical Submersions

Entropy ◽  
2015 ◽  
Vol 17 (12) ◽  
pp. 6213-6228 ◽  
Author(s):  
Alina-Daniela Vîlcu ◽  
Gabriel-Eduard Vîlcu
Keyword(s):  
Statistical Manifolds ◽  
Quaternionic Kähler

scholarly journals Curvature of quaternionic Kähler manifolds with $$S^1$$-symmetry

2021 ◽  
Author(s):  
V. Cortés ◽  
A. Saha ◽  
D. Thung
Keyword(s):  
Curvature Tensor ◽  
Decomposition Theorem ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Cohomogeneity One ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Quaternionic Kähler

AbstractWe study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kähler side in terms of the initial hyper-Kähler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for a series of complete quaternionic Kähler manifolds arising from flat hyper-Kähler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.

scholarly journals Hooke’s law in statistical manifolds and divergences

2000 ◽  
Vol 159 ◽  
pp. 1-24 ◽  
Author(s):  
Masayuki Henmi ◽  
Ryoichi Kobayashi
Keyword(s):  
Classical Mechanics ◽  
Riemannian Metric ◽  
Legendre Transform ◽  
Point Function ◽  
Hooke’S Law ◽  
Affine Connections ◽  
Statistical Manifolds ◽  
Dual Affine ◽  
Hooke's Law

The concept of the canonical divergence is defined for dually flat statistical manifolds in terms of the Legendre transform between dual affine coordinates. In this article, we introduce a new two point function defined for any triple (g,∇, ∇*) of a Riemannian metric g and two affine connections ∇ and ∇*. We show that this interprets the canonical divergence without refering to the existence of special coordinates (dual affine coordinates) but in terms of only classical mechanics concerning ∇- and ∇*-geodesics. We also discuss the properties of the two point function and show that this shares some important properties with the canonical divergence defined on dually flat statistical manifolds.

scholarly journals Manifolds of differentiable densities

2018 ◽  
Vol 22 ◽  
pp. 19-34 ◽  
Author(s):  
Nigel J. Newton
Keyword(s):  
Continuous Function ◽  
Likelihood Function ◽  
Probability Measures ◽  
Fréchet Spaces ◽  
Finite Sample ◽  
Infinite Dimensional ◽  
Statistical Manifolds ◽  
Covariant Derivatives ◽  
Finite Measures ◽  
Non Parametric

We develop a family of infinite-dimensional (non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class Cbk with respect to appropriate reference measures. The case k = ∞, in which the manifolds are modelled on Fréchet spaces, is included. The manifolds admit the Fisher-Rao metric and, unusually for the non-parametric setting, Amari’s α-covariant derivatives for all α ∈ ℝ. By construction, they are C∞-embedded submanifolds of particular manifolds of finite measures. The statistical manifolds are dually (α = ±1) flat, and admit mixture and exponential representations as charts. Their curvatures with respect to the α-covariant derivatives are derived. The likelihood function associated with a finite sample is a continuous function on each of the manifolds, and the α-divergences are of class C∞.

scholarly journals Sasakian statistical manifolds

2017 ◽  
Vol 117 ◽  
pp. 179-186 ◽  
Author(s):  
Hitoshi Furuhata ◽  
Izumi Hasegawa ◽  
Yukihiko Okuyama ◽  
Kimitake Sato ◽  
Mohammad Hasan Shahid
Keyword(s):  
Statistical Manifolds

scholarly journals On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvature

2020 ◽  
Vol 5 (4) ◽  
pp. 3495-3509
Author(s):  
Aliya Naaz Siddiqui ◽  
◽  
Mohammad Hasan Shahid ◽  
Jae Won Lee ◽  
Keyword(s):  
Ricci Curvature ◽  
Constant Curvature ◽  
Statistical Manifolds
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