On the normal scalar curvature conjecture in Kenmotsu statistical manifolds

Warped products play crucial roles in differential geometry, as well as in mathematical physics, especially in general relativity. In this article, first we define and study statistical solitons on Ricci-symmetric statistical warped products R × f N 2 and N 1 × f R . Second, we study statistical warped products as submanifolds of statistical manifolds. For statistical warped products statistically immersed in a statistical manifold of constant curvature, we prove Chen’s inequality involving scalar curvature, the squared mean curvature, and the Laplacian of warping function (with respect to the Levi–Civita connection). At the end, we establish a relationship between the scalar curvature and the Casorati curvatures in terms of the Laplacian of the warping function for statistical warped product submanifolds in the same ambient space.

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Lower Bounds on Statistical Submersions with vertical Casorati curvatures

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s021988782250044x ◽

2021 ◽

Author(s):

Aliya Naaz Siddiqui ◽

Mohd Danish Siddiqi ◽

Ali H. Alkhaldi ◽

Akram Ali

Keyword(s):

Scalar Curvature ◽

Lower Bounds ◽

Statistical Manifolds ◽

Normalized Scalar Curvature ◽

Statistical Submersion

In this paper, we obtain lower bounds for the normalized scalar curvature on statistical submersion with the normalized [Formula: see text]-vertical Casorati curvatures. Also, we discuss the conditions for which the equality cases hold. Beside this, we determine the statistical solitons on statistical submersion from statistical manifolds and illustrate an example of statistical submersions from statistical manifolds.

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Inequalities for the Casorati Curvature of Totally Real Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms

Entropy ◽

10.3390/e23111399 ◽

2021 ◽

Vol 23 (11) ◽

pp. 1399

Author(s):

Bang-Yen Chen ◽

Simona Decu ◽

Gabriel-Eduard Vîlcu

Keyword(s):

Scalar Curvature ◽

Space Form ◽

Space Forms ◽

Statistical Manifolds ◽

Spacelike Submanifolds ◽

Totally Real ◽

Casorati Curvature

The purpose of this article is to establish some inequalities concerning the normalized δ-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of totally real spacelike submanifolds in statistical manifolds of the type para-Kähler space form. Moreover, this study is focused on the equality cases in these inequalities. Some examples are also provided.

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Bounds for Statistical Curvatures of Submanifolds in Kenmotsu-like Statistical Manifolds

Mathematics ◽

10.3390/math10020176 ◽

2022 ◽

Vol 10 (2) ◽

pp. 176

Author(s):

Aliya Naaz Siddiqui ◽

Mohd Danish Siddiqi ◽

Ali Hussain Alkhaldi

Keyword(s):

Scalar Curvature ◽

Fundamental Form ◽

Second Fundamental Form ◽

The Second Fundamental Form ◽

Statistical Manifolds ◽

Equality Condition ◽

Normalized Scalar Curvature ◽

Legendrian Submanifolds

In this article, we obtain certain bounds for statistical curvatures of submanifolds with any codimension of Kenmotsu-like statistical manifolds. In this context, we construct a class of optimum inequalities for submanifolds in Kenmotsu-like statistical manifolds containing the normalized scalar curvature and the generalized normalized Casorati curvatures. We also define the second fundamental form of those submanifolds that satisfy the equality condition. On Legendrian submanifolds of Kenmotsu-like statistical manifolds, we discuss a conjecture for Wintgen inequality. At the end, some immediate geometric consequences are stated.

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Inequalities on Sasakian Statistical Manifolds in Terms of Casorati Curvatures

Mathematics ◽

10.3390/math6110259 ◽

2018 ◽

Vol 6 (11) ◽

pp. 259 ◽

Cited By ~ 2

Author(s):

Chul Lee ◽

Jae Lee

Keyword(s):

Scalar Curvature ◽

Riemannian Metric ◽

Statistical Manifold ◽

Statistical Structure ◽

Slant Submanifolds ◽

Statistical Manifolds ◽

Conjugate Connections ◽

Levi Civita Connection ◽

Normalized Scalar Curvature ◽

Totally Real

A statistical structure is considered as a generalization of a pair of a Riemannian metric and its Levi-Civita connection. With a pair of conjugate connections ∇ and ∇ * in the Sasakian statistical structure, we provide the normalized scalar curvature which is bounded above from Casorati curvatures on C-totally real (Legendrian and slant) submanifolds of a Sasakian statistical manifold of constant φ -sectional curvature. In addition, we give examples to show that the total space is a sphere.

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Curvature Invariants of Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature

Entropy ◽

10.3390/e20070529 ◽

2018 ◽

Vol 20 (7) ◽

pp. 529 ◽

Cited By ~ 11

Author(s):

Simona Decu ◽

Stefan Haesen ◽

Leopold Verstraelen ◽

Gabriel-Eduard Vîlcu

Keyword(s):

Scalar Curvature ◽

Sectional Curvature ◽

Totally Geodesic ◽

Curvature Invariants ◽

Statistical Manifolds ◽

Civita Connection ◽

Levi Civita Connection ◽

Curvature Tensors

In this article, we consider statistical submanifolds of Kenmotsu statistical manifolds of constant ϕ-sectional curvature. For such submanifold, we investigate curvature properties. We establish some inequalities involving the normalized δ-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant). Moreover, we prove that the equality cases of the inequalities hold if and only if the imbedding curvature tensors h and h∗ of the submanifold (associated with the dual connections) satisfy h=−h∗, i.e., the submanifold is totally geodesic with respect to the Levi–Civita connection.

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Inequalities for the Casorati Curvature of Statistical Manifolds in Holomorphic Statistical Manifolds of Constant Holomorphic Curvature

Mathematics ◽

10.3390/math8020251 ◽

2020 ◽

Vol 8 (2) ◽

pp. 251 ◽

Cited By ~ 1

Author(s):

Simona Decu ◽

Stefan Haesen ◽

Leopold Verstraelen

Keyword(s):

Scalar Curvature ◽

Sectional Curvature ◽

Holomorphic Sectional Curvature ◽

Holomorphic Curvature ◽

Statistical Manifolds ◽

Casorati Curvature

In this paper, we prove some inequalities in terms of the normalized δ -Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of statistical submanifolds in holomorphic statistical manifolds with constant holomorphic sectional curvature. Moreover, we study the equality cases of such inequalities. An example on these submanifolds is presented.

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Positive scalar curvature metrics via end-periodic manifolds

Annals of K-Theory ◽

10.2140/akt.2020.5.639 ◽

2020 ◽

Vol 5 (3) ◽

pp. 639-676

Author(s):

Michael Hallam ◽

Varghese Mathai

Keyword(s):

Scalar Curvature ◽

Positive Scalar Curvature ◽

Positive Scalar

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On the quantization of folded strings in non-critical dimensions

Journal of High Energy Physics ◽

10.1007/jhep12(2020)120 ◽

2020 ◽

Vol 2020 (12) ◽

Author(s):

Jacob Sonnenschein ◽

Dorin Weissman

Keyword(s):

Scalar Curvature ◽

Space Dimension ◽

Massive Particle ◽

Open String ◽

Closed String ◽

Target Space ◽

Critical Dimensions ◽

Complete Set ◽

Rotating String ◽

Folding Point

Abstract Classical rotating closed string are folded strings. At the folding points the scalar curvature associated with the induced metric diverges. As a consequence one cannot properly quantize the fluctuations around the classical solution since there is no complete set of normalizable eigenmodes. Furthermore in the non-critical effective string action of Polchinski and Strominger, there is a divergence associated with the folds. We overcome this obstacle by putting a massive particle at each folding point which can be used as a regulator. Using this method we compute the spectrum of quantum fluctuations around the rotating string and the intercept of the leading Regge trajectory. The results we find are that the intercepts are a = 1 and a = 2 for the open and closed string respectively, independent of the target space dimension. We argue that in generic theories with an effective string description, one can expect corrections from finite masses associated with either the endpoints of an open string or the folding points on a closed string. We compute explicitly the corrections in the presence of these masses.

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Quaternionic contact 4n + 3-manifolds and their 4n-quotients

Annals of Global Analysis and Geometry ◽

10.1007/s10455-021-09758-5 ◽

2021 ◽

Author(s):

Yoshinobu Kamishima

Keyword(s):

Scalar Curvature ◽

Lie Group ◽

Contact Structure ◽

Einstein Manifolds ◽

Hermitian Metrics ◽

Kähler Metric ◽

Formula One ◽

Quaternion Space ◽

Kahler Metric ◽

Do So

AbstractWe study some types of qc-Einstein manifolds with zero qc-scalar curvature introduced by S. Ivanov and D. Vassilev. Secondly, we shall construct a family of quaternionic Hermitian metrics $$(g_a,\{J_\alpha \}_{\alpha =1}^3)$$ ( g a , { J α } α = 1 3 ) on the domain Y of the standard quaternion space $${\mathbb {H}}^n$$ H n one of which, say $$(g_a,J_1)$$ ( g a , J 1 ) is a Bochner flat Kähler metric. To do so, we deform conformally the standard quaternionic contact structure on the domain X of the quaternionic Heisenberg Lie group$${{\mathcal {M}}}$$ M to obtain quaternionic Hermitian metrics on the quotient Y of X by $${\mathbb {R}}^3$$ R 3 .

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