Differential Geometry: Structures on Manifolds and Their 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 (30 April 2022) | Viewed by 20657

Special Issue Editor

Faculty of Mathematics, Alexandru Ioan Cuza University of Iasi, Bd. Carol I, No. 11, 700506 Iasi, Romania
Interests: differential geometry; (pseudo-) Riemannian geometry; submanifolds
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Special Issue Information

Dear Colleagues,

When a manifold is endowed with a geometric structure, we have more opportunities to explore its geometric properties. Affine geometry, Riemannian geometry, contact geometry, Kaelher geometry, CR geometry, or Finsler geometry are only a few examples of such differential geometric structures. Several theoretical and practical applications have been obtained over the years: mathematical physics, mathematical biology, economy, and so on. On the other hand, the theory of submanifolds represents an important field in differential geometry, especially when the ambient manifold carries geometric structures. The connection between the intrinsic geometry of the submanifold with its extrinsic geometry has been extensively developed in recent decades.

Prof. Dr. Marian Ioan Munteanu
Guest Editor

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Keywords

  • Contact structures
  • Distributions
  • Geodesic and harmonic maps
  • Delta invariants
  • Minimal submanifolds
  • CR submanifolds
  • Curvature

Published Papers (12 papers)

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Editorial

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3 pages, 158 KiB  
Editorial
Preface to: Differential Geometry: Structures on Manifolds and Their Applications
Mathematics 2022, 10(13), 2243; https://doi.org/10.3390/math10132243 - 27 Jun 2022
Viewed by 914
Abstract
When a manifold is endowed with a geometric structure, we have more opportunities to explore its geometric properties [...] Full article
(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)

Research

Jump to: Editorial

21 pages, 336 KiB  
Article
Bertrand and Mannheim Curves of Spherical Framed Curves in a Three-Dimensional Sphere
Mathematics 2022, 10(8), 1292; https://doi.org/10.3390/math10081292 - 13 Apr 2022
Cited by 1 | Viewed by 1114
Abstract
We investigated differential geometries of Bertrand curves and Mannheim curves in a three-dimensional sphere. We clarify the conditions for regular spherical curves to become Bertrand and Mannheim curves. Then, we concentrate on Bertrand and Mannheim curves of singular spherical curves. As singular spherical [...] Read more.
We investigated differential geometries of Bertrand curves and Mannheim curves in a three-dimensional sphere. We clarify the conditions for regular spherical curves to become Bertrand and Mannheim curves. Then, we concentrate on Bertrand and Mannheim curves of singular spherical curves. As singular spherical curves, we considered spherical framed curves. We define Bertrand and Mannheim curves of spherical framed curves. We give conditions for spherical framed curves to become Bertrand and Mannheim curves. Full article
(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)
18 pages, 359 KiB  
Article
Magnetic Jacobi Fields in 3-Dimensional Cosymplectic Manifolds
Mathematics 2021, 9(24), 3220; https://doi.org/10.3390/math9243220 - 13 Dec 2021
Cited by 4 | Viewed by 1378
Abstract
We classify the magnetic Jacobi fields in cosymplectic manifolds of dimension 3, enriching the results in the study of magnetic Jacobi fields derived from uniform magnetic fields. In particular, we give examples of Jacobi magnetic fields in the Euclidean space E3 and [...] Read more.
We classify the magnetic Jacobi fields in cosymplectic manifolds of dimension 3, enriching the results in the study of magnetic Jacobi fields derived from uniform magnetic fields. In particular, we give examples of Jacobi magnetic fields in the Euclidean space E3 and we conclude with the description of magnetic Jacobi fields in the product spaces S2×R and H2×R. Full article
(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)
12 pages, 273 KiB  
Article
On an Anti-Torqued Vector Field on Riemannian Manifolds
Mathematics 2021, 9(18), 2201; https://doi.org/10.3390/math9182201 - 08 Sep 2021
Cited by 3 | Viewed by 1041
Abstract
A torqued vector field ξ is a torse-forming vector field on a Riemannian manifold that is orthogonal to the dual vector field of the 1-form in the definition of torse-forming vector field. In this paper, we introduce an anti-torqued vector field which is [...] Read more.
A torqued vector field ξ is a torse-forming vector field on a Riemannian manifold that is orthogonal to the dual vector field of the 1-form in the definition of torse-forming vector field. In this paper, we introduce an anti-torqued vector field which is opposite to torqued vector field in the sense it is parallel to the dual vector field to the 1-form in the definition of torse-forming vector fields. It is interesting to note that anti-torqued vector fields do not reduce to concircular vector fields nor to Killing vector fields and thus, give a unique class among the classes of special vector fields on Riemannian manifolds. These vector fields do not exist on compact and simply connected Riemannian manifolds. We use anti-torqued vector fields to find two characterizations of Euclidean spaces. Furthermore, a characterization of an Einstein manifold is obtained using the combination of a torqued vector field and Fischer–Marsden equation. We also find a condition under which the scalar curvature of a compact Riemannian manifold admitting an anti-torqued vector field is strictly negative. Full article
(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)
15 pages, 293 KiB  
Article
Warped Product Submanifolds in Locally Golden Riemannian Manifolds with a Slant Factor
Mathematics 2021, 9(17), 2125; https://doi.org/10.3390/math9172125 - 02 Sep 2021
Cited by 2 | Viewed by 1374
Abstract
In the present paper, we study some properties of warped product pointwise semi-slant and hemi-slant submanifolds in Golden Riemannian manifolds, and we construct examples in Euclidean spaces. Additionally, we study some properties of proper warped product pointwise semi-slant (and, respectively, hemi-slant) submanifolds in [...] Read more.
In the present paper, we study some properties of warped product pointwise semi-slant and hemi-slant submanifolds in Golden Riemannian manifolds, and we construct examples in Euclidean spaces. Additionally, we study some properties of proper warped product pointwise semi-slant (and, respectively, hemi-slant) submanifolds in a locally Golden Riemannian manifold. Full article
(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)
26 pages, 2621 KiB  
Article
Geometric Numerical Integration of Liénard Systems via a Contact Hamiltonian Approach
Mathematics 2021, 9(16), 1960; https://doi.org/10.3390/math9161960 - 16 Aug 2021
Cited by 5 | Viewed by 2260
Abstract
Starting from a contact Hamiltonian description of Liénard systems, we introduce a new family of explicit geometric integrators for these nonlinear dynamical systems. Focusing on the paradigmatic example of the van der Pol oscillator, we demonstrate that these integrators are particularly stable and [...] Read more.
Starting from a contact Hamiltonian description of Liénard systems, we introduce a new family of explicit geometric integrators for these nonlinear dynamical systems. Focusing on the paradigmatic example of the van der Pol oscillator, we demonstrate that these integrators are particularly stable and preserve the qualitative features of the dynamics, even for relatively large values of the time step and in the stiff regime. Full article
(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)
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10 pages, 260 KiB  
Article
Some Conditions on Trans-Sasakian Manifolds to Be Homothetic to Sasakian Manifolds
Mathematics 2021, 9(16), 1887; https://doi.org/10.3390/math9161887 - 08 Aug 2021
Cited by 5 | Viewed by 1319
Abstract
In this paper, we study 3-dimensional compact and connected trans-Sasakian manifolds and find necessary and sufficient conditions under which these manifolds are homothetic to Sasakian manifolds. First, four results in this paper deal with finding necessary and sufficient conditions on a compact and [...] Read more.
In this paper, we study 3-dimensional compact and connected trans-Sasakian manifolds and find necessary and sufficient conditions under which these manifolds are homothetic to Sasakian manifolds. First, four results in this paper deal with finding necessary and sufficient conditions on a compact and connected trans-Sasakian manifold to be homothetic to a compact and connected Sasakian manifold, and the fifth result deals with finding necessary and sufficient condition on a connected trans-Sasakian manifold to be homothetic to a connected Sasakian manifold. Finally, we find necessary and sufficient conditions on a compact and simply connected trans-Sasakian manifold to be homothetic to a compact and simply connected Einstein Sasakian manifold. Full article
(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)
17 pages, 489 KiB  
Article
Principal Bundle Structure of Matrix Manifolds
Mathematics 2021, 9(14), 1669; https://doi.org/10.3390/math9141669 - 15 Jul 2021
Cited by 2 | Viewed by 1421
Abstract
In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold Gr(Rk) of linear subspaces of dimension r<k in [...] Read more.
In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold Gr(Rk) of linear subspaces of dimension r<k in Rk, which avoids the use of equivalence classes. The set Gr(Rk) is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on R(kr)×r. Then, we define an atlas for the set Mr(Rk×r) of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rk) and typical fibre GLr, the general linear group of invertible matrices in Rk×k. Finally, we define an atlas for the set Mr(Rn×m) of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rn)×Gr(Rm) and typical fibre GLr. The atlas of Mr(Rn×m) is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set Mr(Rn×m) equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space Rn×m equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space Rn×m, seen as the union of manifolds Mr(Rn×m), as an analytic manifold equipped with a topology for which the matrix rank is a continuous map. Full article
(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)
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12 pages, 265 KiB  
Article
Representations of Rectifying Isotropic Curves and Their Centrodes in Complex 3-Space
Mathematics 2021, 9(12), 1451; https://doi.org/10.3390/math9121451 - 21 Jun 2021
Cited by 2 | Viewed by 1219
Abstract
In this work, the rectifying isotropic curves are investigated in three-dimensional complex space C3. The conclusion that an isotropic curve is a rectifying curve if and only if its pseudo curvature is a linear function of its pseudo arc-length is achieved. [...] Read more.
In this work, the rectifying isotropic curves are investigated in three-dimensional complex space C3. The conclusion that an isotropic curve is a rectifying curve if and only if its pseudo curvature is a linear function of its pseudo arc-length is achieved. Meanwhile, the rectifying isotropic curves are expressed by the Bessel functions explicitly. Last but not least, the centrodes of rectifying isotropic curves are explored in detail. Full article
(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)
9 pages, 826 KiB  
Article
Geometrical Properties of the Pseudonull Hypersurfaces in Semi-Euclidean 4-Space
Mathematics 2021, 9(11), 1274; https://doi.org/10.3390/math9111274 - 01 Jun 2021
Cited by 5 | Viewed by 1788
Abstract
In this paper, we focus on some geometrical properties of the partially null slant helices in semi-Euclidean 4-space. By structuring suitable height functions, we obtain the singularity types of the pseudonull hypersurfaces, which are generated by the partially null slant helices. An example [...] Read more.
In this paper, we focus on some geometrical properties of the partially null slant helices in semi-Euclidean 4-space. By structuring suitable height functions, we obtain the singularity types of the pseudonull hypersurfaces, which are generated by the partially null slant helices. An example is given to determine the main results. Full article
(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)
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14 pages, 548 KiB  
Article
Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space
Mathematics 2021, 9(11), 1256; https://doi.org/10.3390/math9111256 - 31 May 2021
Cited by 6 | Viewed by 2536
Abstract
In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null curves are spacelike curves with null principal normals, and their involutes can be defined analogously as for the Euclidean curves, but they exhibit properties that cannot occur in Euclidean space. [...] Read more.
In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null curves are spacelike curves with null principal normals, and their involutes can be defined analogously as for the Euclidean curves, but they exhibit properties that cannot occur in Euclidean space. The first result of the paper is that the involutes of pseudo-null curves are null curves, more precisely, null straight lines. Furthermore, a method of reconstruction of a pseudo-null curve from a given null straight line as its involute is provided. Such a reconstruction process in Euclidean plane generates an evolute of a curve, however it cannot be applied to a straight line. In the case presented, the process is additionally affected by a choice of different null frames that every null curve allows (in this case, a null straight line). Nevertheless, we proved that for different null frames, the obtained pseudo-null curves are congruent. Examples that verify presented results are also given. Full article
(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)
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12 pages, 297 KiB  
Article
Canonical Almost Geodesic Mappings of the First Type of Spaces with Affine Connections onto Generalized m-Ricci-Symmetric Spaces
Mathematics 2021, 9(4), 437; https://doi.org/10.3390/math9040437 - 22 Feb 2021
Cited by 10 | Viewed by 1615
Abstract
In the paper we consider almost geodesic mappings of the first type of spaces with affine connections onto generalized 2-Ricci-symmetric spaces, generalized 3-Ricci-symmetric spaces, and generalized m-Ricci-symmetric spaces. In either case the main equations for the mappings are obtained as a closed [...] Read more.
In the paper we consider almost geodesic mappings of the first type of spaces with affine connections onto generalized 2-Ricci-symmetric spaces, generalized 3-Ricci-symmetric spaces, and generalized m-Ricci-symmetric spaces. In either case the main equations for the mappings are obtained as a closed system of linear differential equations of Cauchy type in the covariant derivatives. The obtained results extend an amount of research produced by N.S. Sinyukov, V.E. Berezovski, J. Mikeš. Full article
(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)
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