Differential Geometry and Its Application, 2nd Edition

1. Introduction

With this Editorial, we present a Special Issue of Axioms entitled ‘Differential Geometry and Its Application, 2nd edition’. We launched this Special Edition as a continuation of ‘Differential Geometry and Its Application’. This Special Issue provides a platform showcasing the latest achievements in many branches of theoretical and practical mathematical studies. These relate to Riemannian theories, generalized Riemannian spaces and their mappings. The scope of this Special Issue also includes Finsler geometry, Kenmotsu manifolds, Kaehler manifolds, manifolds with non-symmetric linear connections, cosymplectic manifolds, contact manifolds, statistical manifolds, Minkowski spaces, geodesic mappings, almost geodesic mappings, holomorphically projective mappings, warped products of manifolds, complex space forms, quaternionic space forms, golden manifolds, inequalities, invariants, immersions, etc. Potential authors are encouraged to submit papers that present new ideas in the field of differential geometry, in addition to the above topics. Given the broad scope and widespread interest in this topic, more works should be published in this area.

2. Overview of the Published Papers

This Special Issue contains 17 papers which were accepted for publication after a rigorous reviewing process.

The authors of the first contribution consider dual representations of Bertrand offsets. Surfaces are specified and several new results are gained in terms of their integral invariants. A new description of Bertrand offsets for developable surfaces is given. Furthermore, the authors obtained several relationships through the striction curves of Bertrand offsets of ruled surfaces and their integral invariants.

In the second contribution, the authors find some conditions under which the tangent bundle $TM$ has a dualistic structure. Then, they introduce infinitesimal affine transformations on statistical manifolds and investigate these structures on a special statistical distribution, as well as on a tangent bundle of a statistical manifold. Moreover, they also study the mutual curvatures of a statistical manifold M and its tangent bundle $TM$, investigating their relations. More precisely, the authors obtain the mutual curvatures of well-known connections on the tangent bundle $TM$ (the complete, horizontal, and Sasaki connections) and study their vanishing.

In the third contribution, the authors study several non-Riemannian quantities in Finsler geometry. These non-Riemannian quantities play an important role in understanding the geometric properties of Finsler metrics. In particular, the authors find differential equations of Finsler warped product metrics with vanishing $\xi$-curvature or vanishing H-curvature. Furthermore, they show that, for Finsler warped product metrics, the $\xi$-curvature vanishes if and only if the H-curvature vanishes.

The authors of the fourth contribution investigate the geometrical axioms of Riemannian submersions in the context of the $\eta$-Ricci-Yamabe soliton ($\eta -RY$ soliton) with a potential field. They give the categorization of each fiber of Riemannian submersion as an $\eta -RY$ soliton, an $\eta$-Ricci soliton, and an $\eta$-Yamabe soliton. Additionally, the authors consider the many circumstances under which a target manifold of Riemannian submersion is an $\eta -RY$ soliton, an $\eta$-Ricci soliton, an $\eta$-Yamabe soliton, or a quasi-Yamabe soliton. They deduce a Poisson equation on a Riemannian submersion in a specific scenario if the potential vector field $\omega$ of the soliton is of a gradient type $=:grad\left(\eta \right)$, providing some examples of an $\eta -RY$ soliton to illustrate their findings. Finally, the authors explore a number theoretic approach to Riemannian submersion with totally geodesic fibers.

The aim of the authors of the fifth contribution is to characterize a Riemannian 3-manifold $M^3$ equipped with a semi-symmetric metric $\xi$-connection $\tilde{\nabla }$ with $\rho$-Einstein and gradient $\rho$-Einstein solitons. The existence of a gradient $\rho$-Einstein soliton in an $M^3$ admitting $\tilde{\nabla }$ is ensured by constructing a non-trivial example; in this way, some of the authors’ results are verified. By using the standard tensorial technique, the authors prove that the scalar curvature of $(M^3,\tilde{\nabla })$ satisfies the Poisson equation $\Delta R={\displaystyle \frac{4(2-\sigma -6\rho )}{\rho }}$.

In the sixth contribution, the authors utilize the axode invariants to derive novel hyperbolic proofs of the Euler–Savary and Disteli formulae. The widely recognized inflection circle is situated on the hyperbolic dual unit sphere, in accordance with the principles of the kinematic theory of spherical locomotions. Subsequently, a time-like line congruence is defined and its spatial equivalence is thoroughly studied. The formulated assertions degenerate into a quadratic form, which facilitates a comprehensive understanding of the geometric features of the inflection line congruence.

A principal curve on a surface plays a paramount role in reasonable implementations (contribution seven). A curve on a surface is a principal curve if its tangents are principal directions. Using the Serret–Frenet frame, the surface pencil couple can be expressed as linear combinations of the components of the local frames in Galilean 3-space ${\mathbb{G}}_3$. With these parametric representations, a family of surfaces using principal curves (curvature lines), the authors construct the necessary and sufficient conditions for the given Bertrand couple to be the principal curves on these surfaces. Moreover, they also analyze the necessary and sufficient conditions for the given Bertrand couple to satisfy the principal curves and the geodesic requirements. As implementations of their main conclusions, the authors expand some models to confirm the method.

In the eighth contribution, the authors examine the behavior of the simplest realistic Oregonator model of the BZ-reaction from the perspective of KCC theory. In order to reduce the complexity of the model, the authors initially transformed the first-order differential equation of the Oregonator model into a system of second-order differential equations. Using this approach, the authors describe the evolution of the Oregonator model in geometric terms by considering it as a geodesic in a Finsler space. The authors found five KCC invariants using the general expression of the nonlinear and Berwald connections. To understand the chaotic behavior of the Oregonator model, the deviation vector and its curvature around equilibrium points are studied. The authors then obtain the necessary and sufficient conditions for the parameters of the system in order to achieve Jacobi stability near the equilibrium points. Furthermore, a comprehensive examination was conducted to compare the linear and Jacobi stabilities of the Oregonator model at its equilibrium points; the authors then highlight these instances with a few illustrative examples.

The author of the ninth contribution gives an expository account of differential cohomology and the classification of higher line bundles (also known as $S^1$-banded gerbes) with a connection. He begins by examining how Čech cohomology is used to classify principal bundles and defines their characteristic classes, introducing differential cohomology a la Cheeger and Simons and $S^1$-banded gerbes with a connection.

In the tenth contribution, the authors study isotropic submanifolds in locally metallic product space forms. Firstly, they establish the Chen–Ricci inequality for such submanifolds and determine the conditions under which the inequality becomes equal. Additionally, the authors explore the minimality of Lagrangian submanifolds in locally metallic product space forms, applying the result to create a classification theorem for isotropic submanifolds whose mean curvature is constant. More specifically, they demonstrate that the submanifolds are either a product of two Einstein manifolds with Einstein constants, or they are isometric to a totally geodesic submanifold. The authors provide several examples to support their findings.

In the eleventh contribution, the authors study submanifolds tangent to the Reeb vector field in trans-Sasakian manifolds. They prove Chen’s first inequality and the Chen–Ricci inequality, respectively, for submanifolds in trans-Sasakian manifolds which admit a semi-symmetric, non-metric connection. Moreover, the authors obtain a generalized Euler inequality for special contact slant submanifolds in trans-Sasakian manifolds endowed with a semi-symmetric non-metric connection.

In the twelfth contribution, the authors study and classify left-invariant cross-curvature solitons on Lorentzian three-dimensional Lie groups.

In the thirteenth contribution, the authors explores the Ricci tensor of slant submanifolds within locally metallic product space forms equipped with a semi-symmetric metric connection (SSMC). The authors’ investigation includes a derivation of the Chen–Ricci inequality and an in-depth analysis of its equality. More precisely, if the mean curvature vector at a point vanishes, then the equality case of this inequality is achieved by a unit tangent vector at the point when, and only when, the vector belongs to the normal space. Finally, they have shown that when a point is a totally geodesic point or is totally umbilical with $n=2$, the equality of this inequality holds true for all unit tangent vectors at that point, and vice versa.

In the fourteenth contribution, the authors’ focus revolves around the establishment of a geometric inequality, commonly referred to as Chen’s inequality. They specifically apply this inequality to assess the square norm of the mean curvature vector and the warping function of warped product slant submanifolds. Their investigation takes place within the context of locally metallic product space forms with quarter-symmetric metric connections. Additionally, they delve into the condition that determines when equality is achieved within the inequality. Furthermore, the authors also explore a number of implications of their findings.

The author of contribution fifteen proves that a 2-gerbe has a torsion Dixmier–Douady class if, and only if, the gerbe has locally constant cocycle data. As an example application, the author gives an alternative description of flat twisted vector bundles in terms of locally constant transition maps. These results generalize to n-gerbes for $n=1$ and $n=3$, providing insights into the structure of higher gerbes and their applications in the geometry of twisted vector bundles.

In the sixteenth contribution, the authors define quasi-canonical biholomorphically projective and equitorsion quasi-canonical biholomorphically projective mappings. Some relations between the corresponding curvature tensors of the generalized Riemannian spaces $G{R}_N$ and $G{\overline{R}}_N$ are obtained. At the end, they found the invariant geometric object of an equitorsion quasi-canonical biholomorphically projective mapping.

Finally, in the seventeenth contribution, the authors formulate a data-independent latent space regularization constraint for general unsupervised autoencoders. The regularization relies on sampling the autoencoder Jacobian at Legendre nodes, which are the centers of the Gauss–Legendre quadrature. Revisiting this classic allows the authors to prove that regularized autoencoders ensure a one-to-one re-embedding of the initial data manifold into their latent representation. Demonstrations show that previously proposed regularization strategies, such as contractive autoencoding, cause topological defects even in simple examples, as do convolutional-based (variational) autoencoders. In contrast, topological preservation is ensured by standard multilayer perceptron neural networks when regularized using this approach. This observation extends from the classic FashionMNIST dataset to (low-resolution) MRI brain scans, suggesting that reliable low-dimensional representations of complex high-dimensional datasets can be achieved using this regularization technique.

3. Conclusions

A total of 17 papers were published in this Special Issue, ‘Differential Geometry and Its Application, 2nd edition’. In these works, researchers interested in various aspects of Riemannian space theory and related topics will find interesting insights and inspiring results.