Differentiable Manifolds and Geometric Structures

1. Introduction

This editorial presents 26 research articles published in the Special Issue entitled Differentiable Manifolds and Geometric Structures of the MDPI Mathematics journal, which covers a wide range of topics particularly from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the latest achievements in different areas of differential geometry, among which is counted: the geometry of differentiable manifolds with curvature restrictions such as Golden space forms, Sasakian space forms; diffeological and affine connection spaces; Weingarten and Delaunay surfaces; Chen-type inequalities for submanifolds; statistical submersions; manifolds endowed with different geometric structures (Sasakian, weak nearly Sasakian, weak nearly cosymplectic, LP-Kenmotsu, paraquaternionic); solitons (almost Ricci solitons, almost Ricci–Bourguignon solitons, gradient r-almost Newton–Ricci–Yamabe solitons, statistical solitons, solitons with semi-symmetric connections); vector fields (projective, conformal, Killing, 2-Killing).

2. Overview of the Published Papers

Recent Developments on the First Chen Inequality in Differential Geometry (by Bang-Yen Chen and Gabriel-Eduard Vîlcu) explores advancements in the study of the first Chen inequality, which is a key result in differential geometry related to the geometry of submanifolds in Riemannian spaces. The first Chen inequality provides a relationship between the intrinsic and extrinsic curvatures of a submanifold, offering insights into how curvature properties impact the shape and structure of the manifold. The paper highlights recent progress on the first Chen inequality with a special view towards the $\delta \left(2\right)$-ideal submanifolds, i.e., those submanifolds for which the equality case holds, presenting also some generalizations of Chen’s original inequality involving natural extensions of the first $\delta$-invariant, $\delta \left(2\right)$, for spaces with different additional structures. The survey contains an extensive list of references and proposes some interesting open problems, which motivate the reader to continue the study of new similar invariants in various geometric contexts.

Norden Golden Manifolds with Constant Sectional Curvature and Their Submanifolds (by Fulya Şahin, Bayram Şahin and Feyza Esra Erdoğan) investigates the properties of Norden golden manifolds. The authors introduce a new notion of sectional curvature, namely, the Norden golden sectional curvature and the corresponding notion of Norden golden Ricci curvature and they determine the curvature tensor field of a Norden golden manifold if this new sectional curvature is constant, i.e., if the manifold is a Norden golden space form. Within this context, they find some properties of a semi-invariant submanifold of a Norden golden space form.

Vertices of Ovals with Constant Width Relative to Particular Circles (by Adel Al-rabtah and Kamal Al-Banawi) focuses on the geometric properties of ovals of constant width in relation to specific reference circles. The study investigates the positions and characteristics of the vertices of such ovals considered in the context of particular circles referring to specific centers or radii that influence the geometry of the oval. The authors also provide a new proof of Barbier’s theorem, and complete the theoretical part by some simulation results.

Weak Nearly Sasakian and Weak Nearly Cosymplectic Manifolds (by Vladimir Rovenski) introduces new geometric structures, namely, weak nearly Sasakian, weak nearly cosymplectic and weak nearly Kähler structures. The author characterizes the weak nearly Sasakian and weak nearly cosymplectic hypersurfaces in weak nearly Kähler manifolds, and shows that a weak nearly cosymplectic manifold with parallel Reeb vector field is locally a Riemannian product.

Solitons Equipped with a Semi-Symmetric Metric Connection with Some Applications on Number Theory (by Ali H. Hakami, Mohd. Danish Siddiqi, Aliya Naaz Siddiqui and Kamran Ahmad) investigates some properties of $\eta$-Ricci solitons and gradient $\eta$-Ricci solitons on ($\epsilon$)-Kenmotsu manifolds endowed with a semi-symmetric metric connection having certain curvature restrictions. Moreover, the authors deduce some topological consequences of the existence of a semi-symmetric metric connection on an ($\epsilon$)-Kenmotsu manifold.

Categorical Join and Generating Families in Diffeological Spaces (by E. Macías and R. Mehrabi) focuses on the study of diffeological spaces, a generalization of smooth manifolds that allows for more flexible topological structures. Diffeology provides a framework for dealing with spaces where smooth maps are defined differently from the traditional smooth manifold approach, allowing for a broader class of spaces to be considered within differential topology. The paper provides a categorical interpretation of a generating family, namely, by proving that the diffeological space is the join of the family, i.e., the push-out of the pull-back. The authors emphasize many examples related to finite-dimensional manifolds.

On Some Weingarten Surfaces in the Special Linear Group $SL(2,\mathbb{R})$ (by Marian Ioan Munteanu) investigates the geometry of Weingarten surfaces, classifying Weingarten conoids within the special linear group $SL(2,\mathbb{R})$. Weingarten surfaces are surfaces for which there is a functional relation between their principal curvatures. The author shows that there exist no linear Weingarten nontrivial such conoid and that the only conoids in $SL(2,\mathbb{R})$ with constant Gaussian curvature are the flat ones, and he proves that the invariant surfaces by the left action of the nilpotent group are Weingarten surfaces.

Bounds for Eigenvalues of q-Laplacian on Contact Submanifolds of Sasakian Space Forms (by Yanlin Li, Fatemah Mofarreh, Abimbola Abolarinwa, Norah Alshehri and Akram Ali) focuses on spectral properties of the q-Laplacian operator on contact slant submanifolds of Sasakian space forms. The paper provides upper bounds for the first non-zero eigenvalue of the q-Laplacian on special contact slant submanifolds in Sasakian space forms, generalizing also a Reilly-type inequality within this context.

Eigenvectors of the De-Rham Operator (by Nasser Bin Turki, Sharief Deshmukh and Gabriel-Eduard Vîlcu) provides characterizations of the Euclidean sphere and of the Euclidean space by means of a nonzero eigenvector of the de-Rham operator. The authors show that a compact and connected k-dimensional Riemannian manifold possesses a nonzero vector field which is an eigenvector of the de-Rham operator with nonzero eigenvalue and satisfying an integral inequality if and only if the manifold is isometric to a k-sphere, and that a complete and connected k-dimensional Riemannian manifold is isometric to the k-Euclidean space if and only if there exists a nonzero de-Rham harmonic vector field satisfying certain conditions.

Killing and 2-Killing Vector Fields on Doubly Warped Products (by Adara M. Blaga and Cihan Özgür) explores some properties of Killing vector fields and 2-Killing vector fields on doubly warped product manifolds. The authors determine the relations for a vector field on a doubly warped product and its components on the factor manifolds to be Killing or 2-Killing and relate them to Ricci solitons and Einstein manifolds. They find the condition for a factor manifold to be isometric to the Euclidean space and provide necessary and sufficient conditions for a doubly warped product manifold to be trivial. As physical applications, they describe the 2-Killing vector fields on the standard static spacetime and on the generalized Robertson–Walker spacetime.

Lifts of a Semi-Symmetric Metric Connection from Sasakian Statistical Manifolds to Tangent Bundle (by Rajesh Kumar, Sameh Shenawy, Nasser Bin Turki, Lalnunenga Colney and Uday Chand De) focuses on the complete lifts of semi-symmetric metric connections from Sasakian statistical manifolds to their tangent bundles. More precisely, the authors determine the complete lifts of a semi-symmetric metric connection and of its dual connection from a Sasakian statistical manifold to its tangent bundle, find the relations between their curvatures, Ricci curvatures and scalar curvatures.

Statistical Solitonic Impact on Submanifolds of Kenmotsu Statistical Manifolds (by Abdullah Ali H. Ahmadini, Mohd. Danish Siddiqi and Aliya Naaz Siddiqui) explores the properties of statistical solitons on submanifolds of Kenmotsu statistical manifolds, proving that the submanifold is an $\eta$-Einstein or an Einstein manifold according as the Reeb vector field lies in its tangent or normal bundle, respectively. The authors consider also statistical solitons and almost quasi-Yamabe solitons having as potential vector fields the tangential component of a concircular vector field on the ambient Kenmotsu statistical manifold.

On the Structure of $SO\left(3\right)$: Trace and Canonical Decompositions (by Demeter Krupka and Ján Brajerčík) concerns with the special orthogonal group $SO\left(3\right)$, focusing on the properties of trace mapping and canonical decompositions. As a main result, the authors show that every element of $SO\left(3\right)$ decomposes into a product of three elementary special orthogonal matrices.

Parameterizations of Delaunay Surfaces from Scratch (by Clementina D. Mladenova and Ivaïlo M. Mladenov) explores new parameterizations for Delaunay surfaces of revolution depending on two real parameters with precise geometric meanings, the first one giving information concerning the size of the surface, the second one specifying its shape. The authors analytically describe all the non-trivial Delaunay surfaces of first kind (which contain also the spheres) and of second kind (to which belong the catenoids), characterizing them globally by the coefficients of the first and the second fundamental forms.

The de Rham Cohomology Classes of Hemi-Slant Submanifolds in Locally Product Riemannian Manifolds (by Mustafa Gök and Erol Kiliç) investigates the de-Rham cohomology classes of hemi-slant submanifolds in locally product Riemannian manifolds. The authors characterize the geodesically invariance of the slant and of the anti-invariant distributions and establish sufficient conditions for the existence of a canonical de-Rham cohomology class if the submanifold is proper and closed.

Invariants for Second Type Almost Geodesic Mappings of Symmetric Affine Connection Space (by Nenad O. Vesić, Dušan J. Simjanović and Branislav M. Randjelović) focuses on the study of second type almost geodesic mappings of symmetric affine connection spaces and Riemannian space in Eisenhart’s sense, and their associated invariants. Particularly, the authors obtain one invariant of Thomas-type and two invariants of Weyl-type for these mappings.

Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications (by Hanan Alohali, Sharief Deshmukh, Bang-Yen Chen and Hemangi Madhusudan Shah) provides new characterization results for Euclidean spheres by means of the Hodge vector and the Hodge potential from the Hodge decomposition of a conformal vector field, and by its affinity tensor on a compact and connected Riemannian manifold.

Investigating Statistical Submersions and Their Properties (by Aliya Naaz Siddiqui and Fatimah Alghamdi) examines statistical submersions on a statistical manifold with isometric fibers. The authors prove that if a statistical manifold has a non-trivial statistical submersion with isometric fibers, then it can not be isometrically immersed as a doubly minimal submanifold into any statistical manifold of non-positive sectional curvature, provided that the embedding curvature tensors corresponding to the conjugate connections on the ambient manifold coincide, and it can not be isometrically immersed as a doubly totally geodesic submanifold into any statistical manifold of non-positive sectional curvature.

Projective Vector Fields on Semi-Riemannian Manifolds (by Norah Alshehri and Mohammed Guediri) investigates the properties of projective vector fields in the setting of semi-Riemannian manifolds. The authors provide characterizations for projective vector fields and find conditions for a projective vector field to be a parallel, geodesic, Killing, homothetic or conformal vector field. They prove that a projective vector field which is also conformal must be homothetic, and they deduce that a complete semi-Riemannian manifold which admits a non-Killing, projective and conformal vector field is locally isometric to the Euclidean space.

On LP-Kenmotsu Manifold with Regard to Generalized Symmetric Metric Connection of Type $(\alpha ,\beta )$ (by Doddabhadrappla Gowda Prakasha, Nasser Bin Turki, Mathad Veerabhadraswamy Deepika and İnan Ünal) examines some geometric properties of Lorentzian para-Kenmotsu manifolds with a generalized symmetric metric connection of type $(\alpha ,\beta )$. The authors prove that the locally symmetric, Ricci semi-symmetric or $\phi$-Ricci symmetric conditions imply that the manifold is an Einstein manifold with respect to this connection. Moreover, they obtain some conclusions when the LP-Kenmotsu manifold is projectively flat, projectively semi-symmetric or $\phi$-projectively flat with respect to this connection.

Some Properties of the Potential Field of an Almost Ricci Soliton (by Adara M. Blaga and Sharief Deshmukh) explores some properties of the potential vector field of an almost Ricci soliton. The authors determine necessary and sufficient conditions that must be satisfied by the scalar and the Ricci curvatures for a compact and connected manifold admitting a nontrivial almost Ricci soliton to be isometric to a sphere. On the other hand, if the scalar curvature satisfies a certain inequality and the affinity tensor of the potential vector field vanishes, they find that the soliton must be trivial.

Rigidity and Triviality of Gradient r-Almost Newton–Ricci–Yamabe Solitons (by Mohd Danish Siddiqi and Fatemah Mofarreh) investigates some geometric properties of gradient r-almost Newton–Ricci–Yamabe solitons, focusing on the rigidity and triviality aspects. The authors provide conditions for a closed gradient r-Newton–Ricci–Yamabe soliton immersed into a Riemannian space form to be trivial and obtain a Schur-type inequality in this case. Moreover, they find conditions for a complete r-almost Newton–Ricci–Yamabe soliton immersed into a Riemannian manifold of constant sectional curvature to be minimal or totally geodesic, or to be isometric to a Euclidean space or to a sphere.

Twistor and Reflector Spaces for Paraquaternionic Contact Manifolds (by Stefan Ivanov, Ivan Minchev and Marina Tchomakova) focuses on the properties of twistor and reflector spaces of paraquaternionic contact manifolds. The authors prove that the twistor space of any paraquaternionic contact manifold possesses a natural integrable CR structure, while its reflector space carries a natural integrable para-CR structure, of neutral signatures.

Infinite Dimensional Maximal Torus Revisited (by Mohamed Lemine H. Bouleryah, Akram Ali and Piscoran Laurian-Ioan) investigates the concept of infinite-dimensional torus. The authors show that the infinite-dimensional torus of a sphere is induced by the maximal torus of the unitary group $U\left(m\right)$. As a central result, they prove that the limit of the Weyl group of the finite-dimensional torus $T^m$ consists of the Weyl group of the infinite-dimensional torus.

Magnetic Curves in Homothetic s-th Sasakian Manifolds (by Şaban Güvenç and Cihan Özgür) explores geometric properties of normal magnetic curves within the context of homothetic s-th Sasakian manifolds. The authors prove that the necessary and sufficient condition for a curve in a homothetic s-th Sasakian manifold to be a normal magnetic curve is to belong to a class of ${\theta }_i$-slant helices with osculating order $\le 3$. Moreover, they parameterize the normal magnetic curves in ${\mathbb{R}}^{2n+s}$ which satisfy the Lorentz equation.

Ricci–Bourguignon Almost Solitons with Vertical Torse-Forming Potential on Almost Contact Complex Riemannian Manifolds (by Mancho Manev) delves into a class of manifolds equipped with geometric structures, describing some properties of Ricci–Bourguignon-like almost solitons on almost contact B-metric manifolds, when the potential field of the soliton is torse-forming with respect to the two Levi-Civita connections of the pair of B-metrics and pointwise collinear with the Reeb vector field of the structure. The author computes the Ricci tensors and the scalar curvatures and he deduces some relations between certain special classes of almost contact complex Riemannian manifolds.

3. Conclusions

A number of 26 papers have been published in the Special Issue Differentiable Manifolds and Geometric Structures. In these papers, researchers interested in various aspects of Riemannian manifolds’ theory and other related topics could find interesting insights and inspiring results.