scholarly journals Relations between Extrinsic and Intrinsic Invariants of Statistical Submanifolds in Sasaki-Like Statistical Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1285
Author(s):  
Hülya Aytimur ◽  
Adela Mihai ◽  
Cihan Özgür
Keyword(s):  
Curvature Condition ◽  
Statistical Manifolds ◽  
Chen Inequality

The Chen first inequality and a Chen inequality for the δ(2,2)-invariant on statistical submanifolds of Sasaki-like statistical manifolds, under a curvature condition, are obtained.

scholarly journals Chen Inequalities for Statistical Submanifolds of Kähler-Like Statistical Manifolds

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1202 ◽  
Author(s):  
Hülya Aytimur ◽  
Mayuko Kon ◽  
Adela Mihai ◽  
Cihan Özgür ◽  
Kazuhiko Takano
Keyword(s):  
Curvature Tensor ◽  
Natural Condition ◽  
Tensor Field ◽  
Statistical Manifolds ◽  
Chen Inequality

We consider Kähler-like statistical manifolds, whose curvature tensor field satisfies a natural condition. For their statistical submanifolds, we prove a Chen first inequality and a Chen inequality for the invariant δ ( 2 , 2 ) .

Some inequalities for statistical submanifolds of quaternion Kaehler-like statistical space forms

2019 ◽  
Vol 16 (08) ◽  
pp. 1950129 ◽  
Author(s):  
Mohd. Aquib
Keyword(s):  
Space Forms ◽  
Statistical Manifolds ◽  
Statistical Submanifold ◽  
Totally Real ◽  
Statistical Version ◽  
Chen Inequality ◽  
Statistical Space

Motivated by one of the problems proposed by [Vilcu and Vilcu, Statistical manifolds with almost quaternionic structures and quaternionic Kaehler-like statistical submersions, Entropy 17 (2015) 6213–6228] in this paper, we study the statistical submanifolds of quaternion Kaehler-like statistical space forms and provide an answer to the problem. Further, we derive the statistical version of Chen inequality for totally real statistical submanifold in such ambient.

Chen inequalities on lightlike hypersurfaces of a GRW spacetime

2021 ◽  
Author(s):  
Nergi̇z (Önen) Poyraz
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Lightlike Hypersurfaces ◽  
Chen Inequality

In this paper, we introduce [Formula: see text]-Ricci curvature and [Formula: see text]-scalar curvature on lightlike hypersurfaces of a GRW spacetime. Using these curvatures, we establish some inequalities for lightlike hypersurfaces of a GRW spacetime. Using these inequalities, we obtain some characterizations on lightlike hypersurfaces. We also get Chen–Ricci inequality and Chen inequality on a screen homothetic lightlike hypersurfaces of a GRW spacetime.

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 CLASSICAL N=1 and N=2 SUPER W ALGEBRAS FROM A ZERO-CURVATURE CONDITION

1994 ◽  
Vol 09 (03) ◽  
pp. 383-398 ◽  
Author(s):  
FRANÇOIS GIERES ◽  
STEFAN THEISEN
Keyword(s):  
Superconformal Algebra ◽  
Order System ◽  
Curvature Condition ◽  
First Order ◽  
Zero Curvature

Starting from superdifferential operators in an N=1 superfield formulation, we present a systematic prescription for the derivation of classical N=1 and N=2 super W algebras by imposing a zero-curvature condition on the connection of the corresponding first-order system. We illustrate the procedure on the first nontrivial example (beyond the N=1 superconformal algebra) and also comment on the relation with the Gelfand-Dickey construction of W algebras.

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
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