Submanifolds in Metric Manifolds

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: closed (30 September 2023) | Viewed by 11266

Special Issue Editor

Faculty of Food Engineering, Stefan cel Mare University of Suceava, 720229 Suceava, Romania
Interests: mathematics; statistics

Special Issue Information

Dear Colleagues,

This Special Issue, “Submanifolds in metric manifolds”, will be devoted to the study of an important topic of research in differential geometry, regarding the structure induced on submanifolds by the structure defined on various ambient manifolds.

We call for research articles and review articles focused on issues such as: submanifolds of Riemannian manifolds, submanifolds of Kaehlerian manifolds, submanifolds of Sasakian manifolds, or submanifolds in any types of metric manifolds.

The geometry of any particular submanifolds, such as invariant (or holomorphic) submanifolds; anti-invariant (totally real) submanifolds; semi-invariant submanifolds; slant, semi (or hemi)-slant submanifolds; and warped product (bi-warped, semi- slant or hemi- slant warped product) submanifolds of metric manifolds endowed by structures can be treated in this Special Issue, with examples and applications (characterization properties and inequalities or equalities cases).

Prof. Dr. Cristina-Elena Hretcanu
Guest Editor

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Keywords

  • differentiable manifolds
  • Riemannian manifolds
  • Kaehlerian manifolds
  • Sasakian manifolds
  • metric spaces with differentiable structure
  • invariant (or holomorphic) submanifolds
  • anti-invariant (totally real) submanifolds
  • semi-invariant submanifolds
  • slant, semi (or hemi)-slant submanifolds
  • warped product (bi-warped, semi-slant or hemi- slant warped product) submanifolds
  • totally geodesic, totally umbilical or minimal submanifolds

Published Papers (11 papers)

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Research

16 pages, 305 KiB  
Article
Lightlike Hypersurfaces of Meta-Golden Semi-Riemannian Manifolds
by Feyza Esra Erdoğan, Selcen Yüksel Perktaş, Şerife Nur Bozdağ and Bilal Eftal Acet
Mathematics 2023, 11(23), 4798; https://0-doi-org.brum.beds.ac.uk/10.3390/math11234798 - 28 Nov 2023
Viewed by 603
Abstract
In this research, we embark on the examination of lightlike hypersurfaces within an almost meta-Golden semi-Riemannian manifold. We investigate the properties of the induced structure on a lightlike hypersurface by meta-Golden semi-Riemannian structure. Then, we introduce invariant lightlike hypersurfaces, anti-invariant lightlike hypersurfaces and [...] Read more.
In this research, we embark on the examination of lightlike hypersurfaces within an almost meta-Golden semi-Riemannian manifold. We investigate the properties of the induced structure on a lightlike hypersurface by meta-Golden semi-Riemannian structure. Then, we introduce invariant lightlike hypersurfaces, anti-invariant lightlike hypersurfaces and screen semi-invariant lightlike hypersurfaces of almost meta-Golden semi-Riemannian manifolds and give examples. Full article
(This article belongs to the Special Issue Submanifolds in Metric Manifolds)
20 pages, 371 KiB  
Article
Quasi-Statistical Schouten–van Kampen Connections on the Tangent Bundle
by Simona-Luiza Druta-Romaniuc
Mathematics 2023, 11(22), 4614; https://0-doi-org.brum.beds.ac.uk/10.3390/math11224614 - 10 Nov 2023
Viewed by 537
Abstract
We determine the general natural metrics G on the total space TM of the tangent bundle of a Riemannian manifold (M,g) such that the Schouten–van Kampen connection ¯ associated to the Levi-Civita connection of G is (quasi-)statistical. [...] Read more.
We determine the general natural metrics G on the total space TM of the tangent bundle of a Riemannian manifold (M,g) such that the Schouten–van Kampen connection ¯ associated to the Levi-Civita connection of G is (quasi-)statistical. We prove that the base manifold must be a space form and in particular, when G is a natural diagonal metric, (M,g) must be locally flat. We prove that there exist one family of natural diagonal metrics and two families of proper general natural metrics such that (TM,¯,G) is a statistical manifold and one family of proper general natural metrics such that (TM{0},¯,G) is a quasi-statistical manifold. Full article
(This article belongs to the Special Issue Submanifolds in Metric Manifolds)
20 pages, 777 KiB  
Article
Holonomic and Non-Holonomic Geometric Models Associated to the Gibbs–Helmholtz Equation
by Cristina-Liliana Pripoae, Iulia-Elena Hirica, Gabriel-Teodor Pripoae and Vasile Preda
Mathematics 2023, 11(18), 3934; https://0-doi-org.brum.beds.ac.uk/10.3390/math11183934 - 16 Sep 2023
Viewed by 631
Abstract
By replacing the internal energy with the free energy, as coordinates in a “space of observables”, we slightly modify (the known three) non-holonomic geometrizations from Udriste’s et al. work. The coefficients of the curvature tensor field, of the Ricci tensor field, and of [...] Read more.
By replacing the internal energy with the free energy, as coordinates in a “space of observables”, we slightly modify (the known three) non-holonomic geometrizations from Udriste’s et al. work. The coefficients of the curvature tensor field, of the Ricci tensor field, and of the scalar curvature function still remain rational functions. In addition, we define and study a new holonomic Riemannian geometric model associated, in a canonical way, to the Gibbs–Helmholtz equation from Classical Thermodynamics. Using a specific coordinate system, we define a parameterized hypersurface in R4 as the “graph” of the entropy function. The main geometric invariants of this hypersurface are determined and some of their properties are derived. Using this geometrization, we characterize the equivalence between the Gibbs–Helmholtz entropy and the Boltzmann–Gibbs–Shannon, Tsallis, and Kaniadakis entropies, respectively, by means of three stochastic integral equations. We prove that some specific (infinite) families of normal probability distributions are solutions for these equations. This particular case offers a glimpse of the more general “equivalence problem” between classical entropy and statistical entropy. Full article
(This article belongs to the Special Issue Submanifolds in Metric Manifolds)
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13 pages, 324 KiB  
Article
The (α,p)-Golden Metric Manifolds and Their Submanifolds
by Cristina E. Hretcanu and Mircea Crasmareanu
Mathematics 2023, 11(14), 3046; https://0-doi-org.brum.beds.ac.uk/10.3390/math11143046 - 10 Jul 2023
Viewed by 477
Abstract
The notion of a golden structure was introduced 15 years ago by the present authors and has been a constant interest of several geometers. Now, we propose a new generalization apart from that called the metallic structure, which is also considered by the [...] Read more.
The notion of a golden structure was introduced 15 years ago by the present authors and has been a constant interest of several geometers. Now, we propose a new generalization apart from that called the metallic structure, which is also considered by the authors. By adding a compatible Riemannian metric, we focus on the study of the structure induced on submanifolds in this setting and its properties. Also, to illustrate our results, some suitable examples of this type of manifold are presented. Full article
(This article belongs to the Special Issue Submanifolds in Metric Manifolds)
19 pages, 304 KiB  
Article
On Nearly Sasakian and Nearly Kähler Statistical Manifolds
by Siraj Uddin, Esmaeil Peyghan, Leila Nourmohammadifar and Rawan Bossly
Mathematics 2023, 11(12), 2644; https://0-doi-org.brum.beds.ac.uk/10.3390/math11122644 - 09 Jun 2023
Viewed by 813
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 [...] Read more.
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)
16 pages, 309 KiB  
Article
First Natural Connection on Riemannian Π-Manifolds
by Hristo Manev
Mathematics 2023, 11(5), 1146; https://0-doi-org.brum.beds.ac.uk/10.3390/math11051146 - 25 Feb 2023
Cited by 2 | Viewed by 1010
Abstract
A natural connection with torsion is defined, and it is called the first natural connection on the Riemannian Π-manifold. Relations between the introduced connection and the Levi–Civita connection are obtained. Additionally, relations between their respective curvature tensors, torsion tensors, Ricci tensors, and scalar [...] Read more.
A natural connection with torsion is defined, and it is called the first natural connection on the Riemannian Π-manifold. Relations between the introduced connection and the Levi–Civita connection are obtained. Additionally, relations between their respective curvature tensors, torsion tensors, Ricci tensors, and scalar curvatures in the main classes of a classification of Riemannian Π-manifolds are presented. An explicit example of dimension five is provided. Full article
(This article belongs to the Special Issue Submanifolds in Metric Manifolds)
14 pages, 285 KiB  
Article
The ∗-Ricci Operator on Hopf Real Hypersurfaces in the Complex Quadric
by Rongsheng Ma and Donghe Pei
Mathematics 2023, 11(1), 90; https://0-doi-org.brum.beds.ac.uk/10.3390/math11010090 - 26 Dec 2022
Viewed by 929
Abstract
We study the ∗-Ricci operator on Hopf real hypersurfaces in the complex quadric. We prove that for Hopf real hypersurfaces in the complex quadric, the ∗-Ricci tensor is symmetric if and only if the unit normal vector field is singular. In the following, [...] Read more.
We study the ∗-Ricci operator on Hopf real hypersurfaces in the complex quadric. We prove that for Hopf real hypersurfaces in the complex quadric, the ∗-Ricci tensor is symmetric if and only if the unit normal vector field is singular. In the following, we obtain that if the ∗-Ricci tensor of Hopf real hypersurfaces in the complex quadric is symmetric, then the ∗-Ricci operator is both Reeb-flow-invariant and Reeb-parallel. As the correspondence to the semi-symmetric Ricci tensor, we give a classification of real hypersurfaces in the complex quadric with the semi-symmetric ∗-Ricci tensor. Full article
(This article belongs to the Special Issue Submanifolds in Metric Manifolds)
10 pages, 284 KiB  
Article
Z-Symmetric Manifolds Admitting Schouten Tensor
by Mohabbat Ali, Abdul Haseeb, Fatemah Mofarreh and Mohd Vasiulla
Mathematics 2022, 10(22), 4293; https://0-doi-org.brum.beds.ac.uk/10.3390/math10224293 - 16 Nov 2022
Cited by 1 | Viewed by 870
Abstract
The paper deals with the study of Z-symmetric manifolds (ZS)n admitting certain cases of Schouten tensor (specifically: Ricci-recurrent, cyclic parallel, Codazzi type and covariantly constant), and investigate some geometric and physical properties of the manifold. Moreover, we also study [...] Read more.
The paper deals with the study of Z-symmetric manifolds (ZS)n admitting certain cases of Schouten tensor (specifically: Ricci-recurrent, cyclic parallel, Codazzi type and covariantly constant), and investigate some geometric and physical properties of the manifold. Moreover, we also study (ZS)4 spacetimes admitting Codazzi type Schouten tensor. Finally, we construct an example of (ZS)4 to verify our result. Full article
(This article belongs to the Special Issue Submanifolds in Metric Manifolds)
17 pages, 320 KiB  
Article
Vanishing Homology of Warped Product Submanifolds in Complex Space Forms and Applications
by Ali H. Alkhaldi, Pişcoran Laurian-Ioan, Izhar Ahmad and Akram Ali
Mathematics 2022, 10(20), 3884; https://0-doi-org.brum.beds.ac.uk/10.3390/math10203884 - 19 Oct 2022
Cited by 1 | Viewed by 848
Abstract
In this paper, we prove the nonexistence of stable integral currents in compact oriented warped product pointwise semi-slant submanifold Mn of a complex space form M˜(4ϵ) under extrinsic conditions which involve the Laplacian, the squared norm gradient [...] Read more.
In this paper, we prove the nonexistence of stable integral currents in compact oriented warped product pointwise semi-slant submanifold Mn of a complex space form M˜(4ϵ) under extrinsic conditions which involve the Laplacian, the squared norm gradient of the warped function, and pointwise slant functions. We show that i-the homology groups of Mn are vanished. As applications of homology groups, we derive new topological sphere theorems for warped product pointwise semi-slant submanifold Mn, in which Mn is homeomorphic to a sphere Sn if n4 and if n=3, then M3 is homotopic to a sphere S3 under the assumption of extrinsic conditions. Moreover, the same results are generalized for CR-warped product submanifolds. Full article
(This article belongs to the Special Issue Submanifolds in Metric Manifolds)
13 pages, 297 KiB  
Article
On the Topology of Warped Product Pointwise Semi-Slant Submanifolds with Positive Curvature
by Yanlin Li, Ali H. Alkhaldi, Akram Ali and Pişcoran Laurian-Ioan
Mathematics 2021, 9(24), 3156; https://0-doi-org.brum.beds.ac.uk/10.3390/math9243156 - 08 Dec 2021
Cited by 29 | Viewed by 2131
Abstract
In this paper, we obtain some topological characterizations for the warping function of a warped product pointwise semi-slant submanifold of the form Ωn=NTl×fNϕk in a complex projective space [...] Read more.
In this paper, we obtain some topological characterizations for the warping function of a warped product pointwise semi-slant submanifold of the form Ωn=NTl×fNϕk in a complex projective space CP2m(4). Additionally, we will find certain restrictions on the warping function f, Dirichlet energy function E(f), and first non-zero eigenvalue λ1 to prove that stable l-currents do not exist and also that the homology groups have vanished in Ωn. As an application of the non-existence of the stable currents in Ωn, we show that the fundamental group π1(Ωn) is trivial and Ωn is simply connected under the same extrinsic conditions. Further, some similar conclusions are provided for CR-warped product submanifolds. Full article
(This article belongs to the Special Issue Submanifolds in Metric Manifolds)
10 pages, 262 KiB  
Article
Contact-Complex Riemannian Submersions
by Cornelia-Livia Bejan, Şemsi Eken Meriç and Erol Kılıç
Mathematics 2021, 9(23), 2996; https://0-doi-org.brum.beds.ac.uk/10.3390/math9232996 - 23 Nov 2021
Cited by 4 | Viewed by 1009
Abstract
A submersion from an almost contact Riemannian manifold to an almost Hermitian manifold, acting on the horizontal distribution by preserving both the metric and the structure, is, roughly speaking a contact-complex Riemannian submersion. This paper deals mainly with a contact-complex Riemannian submersion from [...] Read more.
A submersion from an almost contact Riemannian manifold to an almost Hermitian manifold, acting on the horizontal distribution by preserving both the metric and the structure, is, roughly speaking a contact-complex Riemannian submersion. This paper deals mainly with a contact-complex Riemannian submersion from an η-Ricci soliton; it studies when the base manifold is Einstein on one side and when the fibres are η-Einstein submanifolds on the other side. Some results concerning the potential are also obtained here. Full article
(This article belongs to the Special Issue Submanifolds in Metric Manifolds)
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