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10 pages, 282 KiB

Open AccessArticle

Weak Nearly Sasakian and Weak Nearly Cosymplectic Manifolds

by Vladimir Rovenski

Mathematics 2023, 11(20), 4377; https://doi.org/10.3390/math11204377 - 21 Oct 2023

Cited by 6 | Viewed by 1625

Weak contact metric structures on a smooth manifold, introduced by V. Rovenski and R. Wolak in 2022, have provided new insight into the theory of classical structures. In this paper, we define new structures of this kind (called weak nearly Sasakian and weak nearly cosymplectic and nearly Kähler structures), study their geometry and give applications to Killing vector fields. We introduce weak nearly Kähler manifolds (generalizing nearly Kähler manifolds), characterize weak nearly Sasakian and weak nearly cosymplectic hypersurfaces in such Riemannian manifolds and prove that a weak nearly cosymplectic manifold with parallel Reeb vector field is locally the Riemannian product of a real line and a weak nearly Kähler manifold. Full article

(This article belongs to the Special Issue Differentiable Manifolds and Geometric Structures)

19 pages, 304 KiB

Open AccessArticle

On Nearly Sasakian and Nearly Kähler Statistical Manifolds

by Siraj Uddin, Esmaeil Peyghan, Leila Nourmohammadifar and Rawan Bossly

Mathematics 2023, 11(12), 2644; https://doi.org/10.3390/math11122644 - 9 Jun 2023

Cited by 5 | Viewed by 1515

Abstract

In this paper, we introduce the notions of nearly Sasakian and nearly Kähler statistical structures with a non-trivial example. The conditions for a real hypersurface in a nearly Kähler statistical manifold to admit a nearly Sasakian statistical structure are given. We also study invariant and anti-invariant statistical submanifolds of nearly Sasakian statistical manifolds. Finally, some conditions under which such a submanifold of a nearly Sasakian statistical manifold is itself a nearly Sasakian statistical manifold are given. Full article

(This article belongs to the Special Issue Submanifolds in Metric Manifolds)

9 pages, 266 KiB

Open AccessArticle

Nearly Sasakian Manifolds of Constant Type

by Aligadzhi Rustanov

Axioms 2022, 11(12), 673; https://doi.org/10.3390/axioms11120673 - 26 Nov 2022

Viewed by 1586

Abstract

The article deals with nearly Sasakian manifolds of a constant type. It is proved that the almost Hermitian structure induced on the integral manifolds of the maximum dimension of the first fundamental distribution of the nearly Sasakian manifold is a nearly Kähler manifold. It is proved that the class of nearly Sasakian manifolds of the zero constant type coincides with the class of Sasakian manifolds. The concept of constancy of the type of an almost contact metric manifold is introduced through its Nijenhuis tensor, and the criterion of constancy of the type of an almost contact metric manifold is proved. The coincidence of both concepts of type constancy for the nearly Sasakian manifold is proved. It is proved that the almost Hermitian structure induced on the integral manifolds of the maximum dimension of the first fundamental distribution of the almost contact metric manifold of the zero constant type is the Hermitian structure. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

12 pages, 286 KiB

Open AccessArticle

Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in S⁶(1)

by Miroslava Antić and Djordje Kocić

Mathematics 2022, 10(13), 2271; https://doi.org/10.3390/math10132271 - 29 Jun 2022

Cited by 16 | Viewed by 1862

Abstract

It is well known that the sphere

S^{6} (1)

admits an almost complex structure J which is nearly Kähler. If M is a hypersurface of an almost Hermitian manifold with a unit normal vector field N, the tangent vector [...] Read more.

It is well known that the sphere

S^{6} (1)

admits an almost complex structure J which is nearly Kähler. If M is a hypersurface of an almost Hermitian manifold with a unit normal vector field N, the tangent vector field

ξ = - J N

is said to be characteristic or the Reeb vector field. The Jacobi operator with respect to

ξ

is called the structure Jacobi operator, and is denoted by

l = R (\cdot, ξ) ξ

, where R is the curvature tensor on M. The study of Riemannian submanifolds in different ambient spaces by means of their Jacobi operators has been highly active in recent years. In particular, many recent results deal with questions around the existence of hypersurfaces with a structure Jacobi operator that satisfies conditions related to their parallelism. In the present paper, we study the parallelism of the structure Jacobi operator of real hypersurfaces in the nearly Kähler sphere

S^{6} (1)

. More precisely, we prove that such real hypersurfaces do not exist. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds: Volume II)

11 pages, 288 KiB

Open AccessArticle

Nearly Cosymplectic Manifolds of Constant Type

by Aligadzhi Rustanov

Axioms 2022, 11(4), 152; https://doi.org/10.3390/axioms11040152 - 25 Mar 2022

Cited by 3 | Viewed by 2769

Abstract

Fundamental identities characterizing a nearly cosymplectic structure and analytical expressions for the first and second structural tensors are obtained in this paper. An identity that is satisfied by the first structural tensor of a nearly cosymplectic structure is proved as well. A contact analog of nearly cosymplectic manifolds’ constancy of type is introduced in this paper. Pointwise constancy conditions of the type of nearly cosymplectic manifolds are obtained. It is proved that for nearly cosymplectic manifolds of dimension greater than three, pointwise constancy of type is equivalent to global constancy of type. A complete classification of nearly cosymplectic manifolds of constant type is obtained. It is also proved that a nearly cosymplectic manifold of dimension less than seven is a proper nearly cosymplectic manifold. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

18 pages, 284 KiB

Open AccessArticle

Almost Complex Surfaces in the Nearly Kähler SL(2,ℝ) × SL(2,ℝ)

by Elsa Ghandour and Luc Vrancken

Mathematics 2020, 8(7), 1160; https://doi.org/10.3390/math8071160 - 15 Jul 2020

Cited by 5 | Viewed by 2016

Abstract

The space

S L (2, R) \times S L (2, R)

The space

S L (2, R) \times S L (2, R)

admits a natural homogeneous pseudo-Riemannian nearly Kähler structure. We investigate almost complex surfaces in this space. In particular, we obtain a complete classification of the totally geodesic almost complex surfaces and of the almost complex surfaces with parallel second fundamental form. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds)

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