Geometry of Manifolds and Applications

1. Introduction

This editorial presents 24 research articles published in the Special Issue entitled Geometry of Manifolds and Applications of the MDPI Mathematics journal, which covers a wide range of topics from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the latest achievements in many branches of theoretical and applied mathematical studies, among which is counted: the geometry of differentiable manifolds with curvature restrictions such as complex space forms, metallic Riemannian space forms, Hessian manifolds of constant Hessian curvature; optimal inequalities for submanifolds, such as generalized Wintgen inequality, inequalities involving $\delta$-invariants; homogeneous spaces and Poisson–Lie groups; the geometry of biharmonic maps; solitons (Ricci solitons, Yamabe solitons, Einstein solitons) in different geometries such as contact and paracontact geometry, complex and metallic Riemannian geometry, statistical and Weyl geometry; perfect fluid spacetimes.

2. Overview of the Published Papers

Differential Geometry of Submanifolds in Complex Space Forms Involving δ-Invariants (by Bang-Yen Chen, Adara M. Blaga and Gabriel-Eduard Vîlcu) presents some of the most recent developments on the geometry of submanifolds isometrically immersed into complex space forms, examining the properties and behavior of certain invariants, particularly $\delta$-invariants, which relate to curvature or other geometric features that characterize the submanifold’s intrinsic geometry. $\delta$-invariants, also called Chen invariants, are often involved in understanding the way a submanifold is embedded in a higher-dimensional space, influencing the curvature and shape of the manifold. They are linked to the second fundamental form which measures how the submanifold bends in the ambient space. The paper provides optimal inequalities involving such $\delta$-invariants, considering also the equality cases which turn to classify different types of submanifolds in complex space forms, including both minimal submanifolds and totally geodesic submanifolds. Many applications of them into different mathematical areas, for e.g., to spectral geometry, to symplectic geometry, to the theory of Riemannian maps, to warped products are presented.

On Minimal Hypersurfaces of a Unit Sphere (by Amira Ishan, Sharief Deshmukh, Ibrahim Al-Dayel and Cihan Özgür) investigates some properties of minimal hypersurfaces of the unit sphere $S^{n+1}$ embedded in the Euclidean space $E^{n+2}$. A minimal hypersurface is a hypersurface which locally minimizes area, i.e., has zero mean curvature. The paper explores conditions under which a minimal hypersurface of the sphere is totally geodesic, and provides a characterization of a Clifford hypersurface within this context.

η-*-Ricci Solitons and Almost co-Kähler Manifolds (by Arpan Sardar, Mohammad Nazrul Islam Khan and Uday Chand De) explores the relationship between a specific class of almost contact metric manifolds called $(k,\mu )$-almost co-Kähler manifolds, and $\eta$-*-Ricci solitons. The authors provide some consequences of the existing of an $\eta$-*-Ricci soliton on a $(k,\mu )$-almost co-Kähler manifold, considering also the case when the soliton is of gradient-type.

The Geometrical Characterizations of the Bertrand Curves of the Null Curves in Semi-Euclidean 4-Space (by Jianguo Sun and Yanping Zhao) investigates a specific class of curves, known as Bertrand curves, within the context of null curves in semi-Euclidean 4-space. Bertrand curves are pairs of curves whose Frenet-Serret normal plane spanned by the normal and second binormal vector of the Frenet frame in corresponding points coincide. The authors obtain some consequences of the existence of such curves for two particular cases, namely, when the second curvature function vanishes or not.

$f(R,T)$-Gravity Model with Perfect Fluid Admitting Einstein Solitons (by Mohd Danish Siddiqi, Sudhakar K. Chaubey and Mohammad Nazrul Islam Khan) explores some geometrical curvature properties and physical features encoded in the pressure and energy density of a perfect fluid admitting Einstein solitons and gradient Einstein solitons, in the framework of $f(R,T)$-gravity. It is known that $f(R,T)$-gravity theory is an extension of general relativity theory, which is supposed be used to describe the universe late-time rapid expansion. In the gradient case, the authors determine the Poisson equation for some particular cases, such as Dark matter era, Stiff matter era, Radiation era, and Dust matter era.

Bounds for Statistical Curvatures of Submanifolds in Kenmotsu-like Statistical Manifolds (by Aliya Naaz Siddiqui, Mohd Danish Siddiqi and Ali Hussain Alkhaldi) deals with the differential geometry of statistical manifolds, specifically those that exhibit Kenmotsu-like structures, and examines the behavior of their statistical curvatures in the context of submanifolds. The authors derive specific bounds for the statistical curvatures of submanifolds within Kenmotsu-like statistical manifolds and they determine some optimal inequalities for these submanifolds in terms of the normalized scalar curvature and the generalized normalized Casorati curvatures, treating also the equality cases. For Legendrian submanifolds of Kenmotsu-like statistical manifolds, they extend the classical DDVV inequality.

Soliton-Type Equations on a Riemannian Manifold (by Nasser Bin Turki, Adara M. Blaga and Sharief Deshmukh) explores certain soliton-type equations on Riemannian manifolds, the so-called generalized gradient Ricci solitons, which include Miao–Tam and Fischer–Marsden equations. Based on Obata’s Theorem, the authors provide necessary and sufficient conditions for a generalized gradient Ricci soliton to be isometric to a sphere. They also characterize the trivial generalized gradient Ricci solitons with unit geodesic potential vector fields.

Para-Ricci-like Solitons with Arbitrary Potential on Para-Sasaki-like Riemannian Π-Manifolds (by Hristo Manev and Mancho Manev) investigates a particular class of geometric solitons called para-Ricci-like solitons with arbitrary potential vector fields in the context of para-Sasaki-like Riemannian $\mathrm{\Pi }$-manifolds. They find the constant value of the scalar curvature of a para-Sasaki-like Riemannian $\mathrm{\Pi }$-manifold, and, in the 3-dimensional case, the compute the sectional curvatures of some special 2-planes, which is constant, too.

Yamabe Solitons on Some Conformal Almost Contact B-Metric Manifolds (by Mancho Manev) explores a class of geometric solitons, known as Yamabe solitons, on a specific type of manifold, conformal almost contact B-metric manifolds. In particular, the author considers the cases when the initial manifold is cosymplectic and Sasaki-like and finds the transformations under which the transformed manifold is a Yamabe soliton having as potential vector field the transformed Reeb vector field.

On Statistical and Semi-Weyl Manifolds Admitting Torsion (by Adara M. Blaga and Antonella Nannicini) provides ways to construct new quasi-statistical structures with the same or conformal metric starting from a quasi-statistical structure. The authors also introduce the notion of quasi-semi-Weyl manifold within the context of manifolds that admit torsion and place these manifolds in relation with quasi-statistical manifolds.

Generalized Wintgen Inequality for Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature (by Aliya Naaz Siddiqui, Ali Hussain Alkhaldi and Lamia Saeed Alqahtani) establishes a generalized Wintgen inequality for statistical submanifolds in Hessian manifolds with constant Hessian curvature. This result extends classical results about the relationship between curvature and submanifold geometry to a new setting, where the submanifolds are statistical in nature, and the ambient manifold has the specific properties of Hessian geometry.

A Discrete Representation of the Second Fundamental Form (by Alfonso Carriazo, Luis M. Fernández and Antonio Ramírez-de-Arellano) presents a combinatorial representation of the second fundamental form, which is a central concept in differential geometry used to describe how a submanifold bends within a higher-dimensional space. Using this representation, the authors characterize the Lagrangian and totally umbilical submanifolds of Kähler manifolds. Moreover, they apply it for the Clifford torus.

Representing Functions in $H^2$ on the Kepler Manifold via WPOAFD Based on the Rational Approximation of Holomorphic Functions (by Zeyuan Song and Zuoren Sun) has as a main objective, to express any holomorphic and square-integrable function on the Kepler manifold as a series using Fourier analysis. Since these function spaces are reproducing kernel Hilbert spaces, the study considers three distinct domains on the Kepler manifold and introduces the weak pre-orthogonal adaptive Fourier decomposition on these domains. Initially, the weak maximal selection principle is demonstrated to select the coefficients of the series. Additionally, a convergence theorem is proved to validate the accuracy of the proposed method.

h-Almost Ricci–Yamabe Solitons in Paracontact Geometry (by Uday Chand De, Mohammad Nazrul Islam Khan and Arpan Sardar) classifies h-almost Ricci–Yamabe solitons within the context of paracontact geometry. Specifically, the authors provide a characterization of para-Kenmotsu manifolds that are h-almost Ricci–Yamabe solitons, as well as 3-dimensional para-Kenmotsu manifolds that are h-almost gradient Ricci–Yamabe solitons. Additionally, they classify para-Sasakian and para-cosymplectic manifolds that admit h-almost Ricci–Yamabe solitons and h-almost gradient Ricci–Yamabe solitons, respectively.

Spheres and Tori as Elliptic Linear Weingarten Surfaces (by Dong-Soo Kim, Young Ho Kim and Jinhua Qian) describes some geometric properties of certain types of surfaces, providing local characterizations of round spheres and tori immersed in the 3-dimensional unit sphere, along with an analysis of the Laplace operator, the spherical Gauss map, and the Gauss map associated with the elliptic linear Weingarten metric.

Some Inequalities of Hardy Type Related to Witten–Laplace Operator on Smooth Metric Measure Spaces (by Yanlin Li, Abimbola Abolarinwa, Ali H. Alkhaldi and Akram Ali) deals with the study of Hardy-type inequalities in the context of smooth metric measure spaces, focusing on the Witten–Laplace operator. This paper extends several Hardy-type integral inequalities to the context of complete, non-compact smooth metric measure spaces, without imposing any geometric restrictions on the potential function. The approach taken emphasizes certain criteria that allow a smooth metric measure space to satisfy Hardy inequalities associated with the Witten p-Laplace operators.

Locally Homogeneous Manifolds Defined by Lie Algebra of Infinitesimal Affine Transformations (by Vladimir A. Popov) explores the study of locally homogeneous manifolds whose geometric properties are governed by the Lie algebra associated with infinitesimal affine transformations. It answers to the question whether an analytic extension of a locally defined analytic affine connection on a locally homogeneous space can be extended analytically to an affine connection on the homogeneous space. More precisely, the author examines whether a Lie subgroup of a simply connected Lie group is closed or non-closed, specifically when the Lie algebra represents the algebra of all analytic infinitesimal affine transformations of an analytic manifold equipped with an analytic affine connection.

Lifts of a Quarter-Symmetric Metric Connection from a Sasakian Manifold to Its Tangent Bundle (by Mohammad Nazrul Islam Khan, Uday Chand De and Ljubica S. Velimirović) studies the relationship between the lifts of a quarter-symmetric metric connection and of the Levi-Civita connection on a Sasakian manifold to its tangent bundle. The authors derive the expressions for the curvature and projective curvature tensors and examine a particular projectively flat and locally symmetry condition.

Geometry of Tangent Poisson–Lie Groups (by Ibrahim Al-Dayel, Foued Aloui and Sharief Deshmukh) delves into the Poisson geometry of Poisson–Lie groups, specifically examining a lifted structure on the tangent bundle of a Poisson–Lie group endowed with a left invariant contravariant pseudo-Riemannian metric. The authors determine the relationship between the contravariant Levi-Civita connections and between the curvatures of the initial and the lifted structure. In the case when the tangent bundle is equipped with the Sánchez de Álvarez Poisson structure, they prove that the initial manifold is a pseudo-Riemannian Poisson–Lie group if and only if the Sánchez de Álvarez tangent Poisson–Lie group is a pseudo-Riemannian Poisson–Lie group, too.

Curvatures on Homogeneous Generalized Matsumoto Space (by M. K. Gupta, Suman Sharma, Fatemah Mofarreh and Sudhakar Kumar Chaubey) explores some curvature properties of a specific class of homogeneous spaces, namely, homogeneous Finsler spaces with a generalized Matsumoto metric. The authors find the expression for the S-curvature and prove that this space has isotropic S-curvature if and only if the S-curvature is zero. On the other hand, they derived the expression for the mean Berwald curvature, and conclude that the isotropic S-curvature implies that the space is weakly Berwald.

Biharmonic Maps on f-Kenmotsu Manifolds with the Schouten–van Kampen Connection (by Hichem El hendi) focuses on the study of biharmonic maps between f-Kenmotsu manifolds, particularly considering the role of the Schouten–van Kampen connection in this context. More precisely, the author determines the necessary and sufficient conditions for a holomorphic map from a Kähler to an f-Kenmotsu manifold to be harmonic or biharmonic, and he characterizes the harmonicity and biharmonicity of the identity map from an f-Kenmotsu manifold to an f-Kenmotsu manifold equipped with the Schouten–van Kampen connection.

An Optimal Inequality for the Normal Scalar Curvature in Metallic Riemannian Space Forms (by Siraj Uddin, Majid Ali Choudhary and Najwa Mohammed Al-Asmari) derives the generalized Wintgen inequality for slant submanifolds of locally metallic space forms equipped with a semi-symmetric metric connection, which relates the normal scalar curvature to the square norm of the mean curvature.

Solitonic View of Generic Contact $CR$-Submanifolds of Sasakian Manifolds with Concurrent Vector Fields (by Vandana, Rajeev Budhiraja, Aliya Naaz Siddiqui and Ali Hussain Alkhaldi) is focused on computing the Ricci tensor when the generic contact $CR$-submanifold of a Sasakian manifold endowed with a concurrent vector field is totally geodesic or when it admits a Ricci or Ricci–Yamabe soliton.

Characterizations of the Frame Bundle Admitting Metallic Structures on Almost Quadratic ϕ-Manifolds (by Mohammad Nazrul Islam Khan, Uday Chand De and Teg Alam) explores the properties of the complete lifts of an almost quadratic $\varphi$-structure to a metallic structure on the tangent bundle. In particular, the authors determine the associated fundamental 2-form and its Nijenhuis tensor field, the vanishing of the latest one being a criterion to establish the integrability of the structure.

3. Conclusions

A number of 24 papers have been published in the Special Issue, Geometry of Manifolds and Applications. In these papers, researchers interested in various aspects of Riemannian manifolds’ theory and other related topics could find interesting insights and inspiring results.