Advances in Differential Geometry and Its Applications

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

Deadline for manuscript submissions: closed (31 August 2024) | Viewed by 13716

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Laboratoire de Mathématiques Appliquées, Ecole Nationale de l'Aviation Civile, 7 Av. Edouard Belin, 31400 Toulouse, France
Interests: information geometry; dual connections; gauge structures; foliations; differential geometry applied to machine learning
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Special Issue Information

Dear Colleagues,

Differential geometry is a wide and active research field in mathematics that has recently received increasing interest from statistics and machine learning. Information geometry, manifold learning, and data manifolds are examples of successful differential geometry uses.

This Special Issue welcomes papers presenting new theoretical results in various areas of differential geometry, as well as applications-oriented articles and aims at a balance between both aspects. Review articles will also be considered.

Dr. Stéphane Puechmorel
Guest Editor

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Keywords

  • Riemannian geometry
  • symplectic geometry
  • contact geometry
  • information geometry
  • gauge theory
  • lie groups and lie algebras
  • applications to machine learning
  • applications to data analysis

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Related Special Issue

Published Papers (13 papers)

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Research

16 pages, 278 KiB  
Article
Characterizations of Spheres and Euclidean Spaces by Conformal Vector Fields
by Sharief Deshmukh, Nasser Bin Turki and Ramesh Sharma
Mathematics 2024, 12(20), 3163; https://doi.org/10.3390/math12203163 - 10 Oct 2024
Viewed by 395
Abstract
A nontrivial conformal vector field ω on an m-dimensional connected Riemannian manifold Mm,g has naturally associated with it the conformal potential θ, a smooth function on Mm, and a skew-symmetric tensor T of type [...] Read more.
A nontrivial conformal vector field ω on an m-dimensional connected Riemannian manifold Mm,g has naturally associated with it the conformal potential θ, a smooth function on Mm, and a skew-symmetric tensor T of type (1,1) called the associated tensor. There is a third entity, namely the vector field Tω, called the orthogonal reflection field, and in this article, we study the impact of the condition that commutator ω,Tω=0; this condition that we refer to as the orthogonal reflection field is commutative. As a natural impact of this condition, we see the existence of a smooth function ρ on Mm such that θ=ρω; this function ρ is called the proportionality function. First, we show that an m-dimensional compact and connected Riemannian manifold Mm,g admits a nontrivial conformal vector field ω with a commuting orthogonal reflection Tω and constant proportionality function ρ if and only if Mm,g is isometric to the sphere Sm(c) of constant curvature c. Secondly, we find three more characterizations of the sphere and two characterizations of a Euclidean space using these ideas. Finally, we provide a condition for a conformal vector field on a compact Riemannian manifold to be closed. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)
12 pages, 371 KiB  
Article
Generalized Bertrand Curves of Non-Light-like Framed Curves in Lorentz–Minkowski 3-Space
by Linlin Wu, Anjie Zhou, Kaixin Yao and Donghe Pei
Mathematics 2024, 12(16), 2593; https://doi.org/10.3390/math12162593 - 22 Aug 2024
Viewed by 558
Abstract
In this paper, we define the generalized Bertrand curves of non-light-like framed curves in Lorentz–Minkowski 3-space; their study is essential for understanding many classical and modern physics problems. Here, we consider two non-light-like framed curves as generalized Bertrand pairs. Our generalized Bertrand pairs [...] Read more.
In this paper, we define the generalized Bertrand curves of non-light-like framed curves in Lorentz–Minkowski 3-space; their study is essential for understanding many classical and modern physics problems. Here, we consider two non-light-like framed curves as generalized Bertrand pairs. Our generalized Bertrand pairs can include Bertrand pairs with either singularities or not, and also include Mannheim pairs with singularities. In addition, we discuss their properties and prove the necessary and sufficient conditions for two non-light-like framed curves to be generalized Bertrand pairs. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)
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13 pages, 280 KiB  
Article
On Sequential Warped Products Whose Manifold Admits Gradient Schouten Harmonic Solitons
by Lixu Yan, Yanlin Li, Fatemah Mofarreh, Akram Ali and Pişcoran Laurian-Ioan
Mathematics 2024, 12(16), 2451; https://doi.org/10.3390/math12162451 - 7 Aug 2024
Viewed by 1363
Abstract
As part of our study, we investigate gradient Schouten harmonic solutions to sequential warped product manifolds. The main contribution of our work is an explanation of how it is possible to express gradient Schouten harmonic solitons on sequential warped product manifolds. Our analysis [...] Read more.
As part of our study, we investigate gradient Schouten harmonic solutions to sequential warped product manifolds. The main contribution of our work is an explanation of how it is possible to express gradient Schouten harmonic solitons on sequential warped product manifolds. Our analysis covers both sequential generalized Robertson–Walker spacetimes and sequential static spacetimes using gradient Schouten harmonic solitons. Studies conducted previously can be generalized from this study. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)
21 pages, 331 KiB  
Article
Contravariant Curvatures of Doubly Warped Product Poisson Manifolds
by Foued Aloui, Shyamal Kumar Hui and Ibrahim Al-Dayel
Mathematics 2024, 12(8), 1205; https://doi.org/10.3390/math12081205 - 17 Apr 2024
Viewed by 580
Abstract
In this paper, we investigate the sectional contravariant curvature of a doubly warped product manifold ( fB×bF,g˜,Π=ΠB+ΠF) equipped with a product Poisson structure Π, using [...] Read more.
In this paper, we investigate the sectional contravariant curvature of a doubly warped product manifold ( fB×bF,g˜,Π=ΠB+ΠF) equipped with a product Poisson structure Π, using warping functions and sectional curvatures of its factor manifolds (B,g˜B,ΠB) and (F,g˜F,ΠF). Qualar and null sectional contravariant curvatures of ( fB×bF,g˜,Π) are also given. As an example, we construct a four-dimensional Lorentzian doubly warped product Poisson manifold where qualar and sectional curvatures are obtained. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)
16 pages, 319 KiB  
Article
f-Biharmonic Submanifolds in Space Forms and f-Biharmonic Riemannian Submersions from 3-Manifolds
by Ze-Ping Wang and Li-Hua Qin
Mathematics 2024, 12(8), 1184; https://doi.org/10.3390/math12081184 - 15 Apr 2024
Viewed by 865
Abstract
f-biharmonic maps are generalizations of harmonic maps and biharmonic maps. In this paper, we give some descriptions of f-biharmonic curves in a space form. We also obtain a complete classification of proper f-biharmonic isometric immersions of a developable surface in [...] Read more.
f-biharmonic maps are generalizations of harmonic maps and biharmonic maps. In this paper, we give some descriptions of f-biharmonic curves in a space form. We also obtain a complete classification of proper f-biharmonic isometric immersions of a developable surface in R3 by proving that a proper f-biharmonic developable surface exists only in the case where the surface is a cylinder. Based on this, we show that a proper biharmonic conformal immersion of a developable surface into R3 exists only in the case when the surface is a cylinder. Riemannian submersions can be viewed as a dual notion of isometric immersions (i.e., submanifolds). We also study f-biharmonicity of Riemannian submersions from 3-manifolds by using the integrability data. Examples are given of proper f-biharmonic Riemannian submersions and f-biharmonic surfaces and curves. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)
19 pages, 310 KiB  
Article
The Gauge Equation in Statistical Manifolds: An Approach through Spectral Sequences
by Michel Nguiffo Boyom and Stephane Puechmorel
Mathematics 2024, 12(8), 1177; https://doi.org/10.3390/math12081177 - 14 Apr 2024
Viewed by 1009
Abstract
The gauge equation is a generalization of the conjugacy relation for the Koszul connection to bundle morphisms that are not isomorphisms. The existence of nontrivial solution to this equation, especially when duality is imposed upon related connections, provides important information about the geometry [...] Read more.
The gauge equation is a generalization of the conjugacy relation for the Koszul connection to bundle morphisms that are not isomorphisms. The existence of nontrivial solution to this equation, especially when duality is imposed upon related connections, provides important information about the geometry of the manifolds under consideration. In this article, we use the gauge equation to introduce spectral sequences that are further specialized to Hessian structures. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)
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10 pages, 237 KiB  
Article
Kropina Metrics with Isotropic Scalar Curvature via Navigation Data
by Yongling Ma, Xiaoling Zhang and Mengyuan Zhang
Mathematics 2024, 12(4), 505; https://doi.org/10.3390/math12040505 - 6 Feb 2024
Cited by 1 | Viewed by 829
Abstract
Through an interesting physical perspective and a certain contraction of the Ricci curvature tensor in Finsler geometry, Akbar-Zadeh introduced the concept of scalar curvature for the Finsler metric. In this paper, we show that the Kropina metric is of isotropic scalar curvature if [...] Read more.
Through an interesting physical perspective and a certain contraction of the Ricci curvature tensor in Finsler geometry, Akbar-Zadeh introduced the concept of scalar curvature for the Finsler metric. In this paper, we show that the Kropina metric is of isotropic scalar curvature if and only if F is an Einstein metric according to the navigation data. Moreover, we obtain the three-dimensional rigidity theorem for an Einstein–Kropina metric. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)
17 pages, 349 KiB  
Article
Twisted Hypersurfaces in Euclidean 5-Space
by Yanlin Li and Erhan Güler
Mathematics 2023, 11(22), 4612; https://doi.org/10.3390/math11224612 - 10 Nov 2023
Cited by 15 | Viewed by 1167
Abstract
The twisted hypersurfaces x with the (0,0,0,0,1) rotating axis in five-dimensional Euclidean space E5 is considered. The fundamental forms, the Gauss map, and the shape operator of x are calculated. In [...] Read more.
The twisted hypersurfaces x with the (0,0,0,0,1) rotating axis in five-dimensional Euclidean space E5 is considered. The fundamental forms, the Gauss map, and the shape operator of x are calculated. In E5, describing the curvatures by using the Cayley–Hamilton theorem, the curvatures of hypersurfaces x are obtained. The solutions of differential equations of the curvatures of the hypersurfaces are open problems. The umbilically and minimality conditions to the curvatures of x are determined. Additionally, the Laplace–Beltrami operator relation of x is given. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)
13 pages, 479 KiB  
Article
One-Parameter Hyperbolic Spatial Locomotions and Invariants of the Axode
by Areej A. Almoneef and Rashad A. Abdel-Baky
Mathematics 2023, 11(17), 3749; https://doi.org/10.3390/math11173749 - 31 Aug 2023
Viewed by 820
Abstract
In this paper, based on the E. Study map, direct appearances were sophisticated for one-parameter hyperbolic dual spherical locomotions and invariants of the axodes. With the suggested technique, the Disteli formulae for the axodes were acquired and the correlations through kinematic geometry of [...] Read more.
In this paper, based on the E. Study map, direct appearances were sophisticated for one-parameter hyperbolic dual spherical locomotions and invariants of the axodes. With the suggested technique, the Disteli formulae for the axodes were acquired and the correlations through kinematic geometry of a timelike line trajectory were provided. Then, a ruled analogy of the curvature circle of a curve in planar locomotions was expanded into generic spatial locomotions. Lastly, we present new hyperbolic proofs for the Euler–Savary and Disteli formulae. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)
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12 pages, 453 KiB  
Article
Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space
by Kemeng Liu, Zewen Li and Donghe Pei
Mathematics 2023, 11(17), 3714; https://doi.org/10.3390/math11173714 - 29 Aug 2023
Viewed by 1092
Abstract
In this paper, we study the surfaces of osculating circles, which are the sets of all osculating circles at all points of regular curves. Since the surfaces of osculating circles may be singular, it is necessary to investigate the singular points of these [...] Read more.
In this paper, we study the surfaces of osculating circles, which are the sets of all osculating circles at all points of regular curves. Since the surfaces of osculating circles may be singular, it is necessary to investigate the singular points of these surfaces. However, traditional methods and tools for analyzing singular properties have certain limitations. To solve this problem, we define the framed surfaces of osculating circles in the Euclidean 3-space. Then, we discuss the types of singular points using the theory of framed surfaces and show that generic singular points of the surfaces consist of cuspidal edges and cuspidal cross-caps. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)
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12 pages, 329 KiB  
Article
A Hypersurfaces of Revolution Family in the Five-Dimensional Pseudo-Euclidean Space E25
by Yanlin Li and Erhan Güler
Mathematics 2023, 11(15), 3427; https://doi.org/10.3390/math11153427 - 7 Aug 2023
Cited by 19 | Viewed by 1676
Abstract
We present a family of hypersurfaces of revolution distinguished by four parameters in the five-dimensional pseudo-Euclidean space E25. The matrices corresponding to the fundamental form, Gauss map, and shape operator of this family are computed. By utilizing the Cayley–Hamilton theorem, [...] Read more.
We present a family of hypersurfaces of revolution distinguished by four parameters in the five-dimensional pseudo-Euclidean space E25. The matrices corresponding to the fundamental form, Gauss map, and shape operator of this family are computed. By utilizing the Cayley–Hamilton theorem, we determine the curvatures of the specific family. Furthermore, we establish the criteria for maximality within this framework. Additionally, we reveal the relationship between the Laplace–Beltrami operator of the family and a 5×5 matrix. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)
17 pages, 328 KiB  
Article
Developable Surfaces Foliated by General Ellipses in Euclidean Space R3
by Ahmad T. Ali
Mathematics 2023, 11(14), 3200; https://doi.org/10.3390/math11143200 - 21 Jul 2023
Viewed by 921
Abstract
In this article, we classify the developable surfaces in three-dimensional Euclidean space R3 that are foliated by general ellipses. We show that the surface has constant Gaussian curvature (CGC) and is foliated by general ellipses if and only if the surface is [...] Read more.
In this article, we classify the developable surfaces in three-dimensional Euclidean space R3 that are foliated by general ellipses. We show that the surface has constant Gaussian curvature (CGC) and is foliated by general ellipses if and only if the surface is developable, i.e., the Gaussian curvature G vanishes everywhere. We characterize all developable surfaces foliated by general ellipses. Some of these surfaces are conical surfaces, and the others are surfaces generated by some special base curves. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)
12 pages, 283 KiB  
Article
CDE’ Inequality on Graphs with Unbounded Laplacian
by Desheng Hong and Chao Gong
Mathematics 2023, 11(9), 2138; https://doi.org/10.3390/math11092138 - 3 May 2023
Viewed by 945
Abstract
In this paper, we derive the gradient estimates of semigroups in terms of the modified curvature-dimension inequality CDE for unbounded Laplacians on complete graphs with non-degenerate measures. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)
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