Casorati Curvatures of Submanifolds in Cosymplectic Statistical Space Forms
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Casorati Curvatures of Submanifolds in Cosymplectic Statistical Space Forms Fereshteh Malek1 · Haniyeh Akbari1 Received: 23 April 2019 / Revised: 3 November 2019 / Accepted: 6 November 2019 © Iranian Mathematical Society 2019
Abstract In this paper, we obtain two inequalities in statistical submanifolds in cosymplectic statistical manifolds with constant φ-sectional curvature which contain generalized normalized δ-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariants). We also investigate the equality cases. Keywords Statistical manifold · Cosymplectic statistical manifold · Generalized normalized δ-Casorati curvature · Scalar curvature Mathematics Subject Classification 53C05 · 53C40
1 Introduction In 1985, statistical manifold was introduced by Amari [1] . There are close relationship between the geometry of statistical manifolds with information geometry, affine geometry, and Hessian geometry. Studying statistical manifolds is an attractive field for several authors. Recently, in [9], Furuhata has studied hypersurfaces in statistical manifolds and has proved that a holomorphic statistical hypersurface with constant holomorphic sectional curvature in a statistical space form is a special Kahler manifold. In [10], Furuhata et al. defined Sasakian statistical manifolds and obtained the condition for a hypersurface in a holomorphic statistical manifold, such that the hypersurface has a Sasakian statistical structure. In [11], Furuhata et al. introduced Kenmotsu statistical manifolds and established that a Kenmotsu statistical manifold
Communicated by Fatemeh Helen Ghane.
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Fereshteh Malek [email protected] Haniyeh Akbari [email protected]
1
Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
123
Bulletin of the Iranian Mathematical Society
could be considered as the warped product of a holomorphic statistical manifold and a line, and they also proved that a Kenmotsu statistical space form could be obtained from a special Kahler manifold. To have a geometric understandingof statistical inference, let Ω be a fixed event space, ρ(Ω) = { p : Ω −→ | Ω p(x)dx = 1, p(x) ≥ 0, ∀x ∈ Ω} be its probability distribution, and Θ ⊂ n be the parameter space on an n-dimensional smooth family defined on Ω. Then, M = { p(x, θ ) ∈ ρ(Ω) | θ = (θ 1 , . . . , θ n ) ∈ Θ} with Riemannian metric: ∂ log p(x, θ ) ∂ log p(x, θ ) p(x, θ )dx dθ i dθ j , g= ∂θ i ∂θ j Ω can be considered as a statistical manifold [1]. In 1993, Chen defined the notion of Chen invariants and established some inequalities consisting of intrinsic invariants and extrinsic invariants for Riemannian submanifolds in real space forms which is one of the most fundamental problems in the submanifold theory [6]. In fact, Chen answered the question of whether there are minimal immersions into any Euclidean spaces that was raised by Chern in 1968. Since then, efforts have been made to generalize these kinds of inequalities. In this context, Mihai et al. [2] have studied the curvature
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