A sectional curvature for statistical structures

Opozda, Barbara

doi:10.1016/j.laa.2016.02.021

Cited by 55 publications

(58 citation statements)

References 10 publications

(26 reference statements)

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“…We want to point-out that this definition has the opposite sign that the sectional curvature defined by B. Opozda [23]. Another sectional curvature was considered in [22] (see also [25]).…”

Section: )mentioning

confidence: 99%

Wintgen inequality for statistical surfaces

Aydın¹,

Iordache²

2019

Mathematical Inequalities &Amp; Applications

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The Wintgen inequality (1979) is a sharp geometric inequality for surfaces in the 4-dimensional Euclidean space involving the Gauss curvature (intrinsic invariant) and the normal curvature and squared mean curvature (extrinsic invariants), respectively.In the present paper we obtain a Wintgen inequality for statistical surfaces. For surfaces M 2 of the Euclidean space E 3 , the Euler inequality K ≤ H 2 is fulfilled, where K is the (intrinsic) Gauss curvature of M 2 and H 2 is the (extrinsic) squared mean curvature of M 2 . Furthermore, K = H 2 everywhere on M 2 if and only if M 2 is totally umbilical, or still, by a theorem of Meusnier, if and only if M 2 is (a part of) a plane E 2 or, it is (a part of) a round sphere S 2 in E 3 . In 1979, P. Wintgen [28] proved that the Gauss curvature K, the squared mean curvature H 2 and the normal curvature K ⊥ of any surface M 2 in E 4 always satisfy the inequality K ≤ H 2 − |K ⊥ |; the equality holds if and only if the ellipse of curvature of M 2 in E 4 is a circle. The Whitney 2-sphere satisfies the equality case of the Wintgen inequality identically. A survey containing recent results on surfaces satisfying identically the equality case of Wintgen inequality can be read in [4]. Later, the Wintgen inequality was extended by B. Rouxel [24] and by I.V. Guadalupe and L. Rodriguez [11] independently, for surfaces M 2 of arbitrary codimension m in real space forms M 2+m (c); namely K ≤ H 2 − |K ⊥ | + c.

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“…We want to point-out that this definition has the opposite sign that the sectional curvature defined by B. Opozda [23]. Another sectional curvature was considered in [22] (see also [25]).…”

Section: )mentioning

confidence: 99%

Wintgen inequality for statistical surfaces

Aydın¹,

Iordache²

2019

Mathematical Inequalities &Amp; Applications

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“…In general, the dual connections are not metric; it follows that one cannot define a sectional curvature with respect to them by the standard definition from Riemannian geometry. B. Opozda proposed two different definitions, in [11,12]. We will work in this article with the definition from [12].…”

Section: Preliminariesmentioning

confidence: 99%

“…B. Opozda proposed two different definitions, in [11,12]. We will work in this article with the definition from [12].…”

Section: Preliminariesmentioning

confidence: 99%

Chen Inequalities for Statistical Submanifolds of Kähler-Like Statistical Manifolds

et al. 2019

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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). MSC: 53C05; 53C40where ∇ 0 is the Levi-Civita connection on M.Similar definitions can be considered for semi-Riemannian manifolds (see also [5]).A statistical structure is said to be of constant curvature ∈ R [6] if:for any vector fields X, Y, Z. The same equation holds for R * (X, Y)Z.A statistical structure of null constant curvature is called a Hessian structure. In [2,7], K. Takano considered a (semi-)Riemannian manifold M, g with an almost complex structure J, endowed with another tensor field J * of type (1, 1) satisfying:for vector fields X and Y on M, g . Then, M, g, J is called an almost Hermite-like manifold. It is easy to see that J * * = J, J * 2 = −I and g JX, J * Y = g (X, Y) . If J is parallel with respect to ∇, then M, g, ∇, J is called a Kähler-like statistical manifold [7]. One also has: g

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“…We want to point-out that, generally, the dual connections are not metric; then one cannot define a sectional curvature with respect to them by the standard definitions. However there exists a sectional curvature on a statistical manifold defined by B. Opozda (see [4]). …”

Section: Introductionmentioning

confidence: 99%