Special Issue "Applications of Differential Geometry"

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (31 October 2018)

Special Issue Editor

Guest Editor
Prof. Dr. Anna Maria Fino

Dipartimento di Matematica “Giuseppe Peano”, Università degli Studi di Torino, Turin 8-10124, Italy
Website | E-Mail
Interests: differential geometry; complex geometry; Lie groups; geometric flows

Special Issue Information

Dear Colleagues,

Differential geometry deals with the application of methods of local and global analysis to geometric problems. It was developed during the 18th and 19th century with the the theory of curves and surfaces in the three-dimensional Euvclidean space. From the 19th century it has grown, considering more generally geometric structures on differential manifolds.

It is deeply linked to other areas of mathematics, such as partial differential equations, topology, complex analytic functions, dynamical systems and group theory.

The goal of this Special Issue is to explore the multifaceted realm of differential geometry, providing a collection of research and survey papers that reflect the research in differential geometry and explore applications in other areas.

Indeed, differential geometry is, not only the standard language used to formulate general relativity, but it has found applications also in medical imaging, computer vision, Hamiltonian mechanics, geometrothermodynamics, geometric design, geometric control and information geometry.

We look forward to your contributions to this Special Issue,

Prof. Dr. Anna Maria Fino
Guest Editor

Manuscript Submission Information

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Keywords

  • Riemannian geometry
  • Symplectic geometry
  • Contact geometry
  • Complex geometry
  • Geometric structures
  • Special geometries
  • Submanifold theory
  • Geometric flows
  • Finsler geometry
  • General relativity
  • String theory
  • Medical imaging
  • Computer vision
  • Hamiltonian mechanics
  • Geometrothermodynamics
  • Geometric design
  • Geometric Control
  • Information geometry

Published Papers (9 papers)

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Research

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Open AccessArticle General Relativity with a Positive Cosmological Constant Λ as a Gauge Theory
Received: 16 October 2018 / Revised: 30 January 2019 / Accepted: 4 February 2019 / Published: 21 February 2019
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Abstract
In this paper, we show that the general relativity action (and Lagrangian) in recent Einstein–Palatini formulation is equivalent in four dimensions to the action (and Langrangian) of a gauge field. First, we briefly showcase the Einstein–Palatini (EP) action, and then we present how [...] Read more.
In this paper, we show that the general relativity action (and Lagrangian) in recent Einstein–Palatini formulation is equivalent in four dimensions to the action (and Langrangian) of a gauge field. First, we briefly showcase the Einstein–Palatini (EP) action, and then we present how Einstein fields equations can be derived from it. In the next section, we study Einstein–Palatini action integral for general relativity with a positive cosmological constant Λ in terms of the corrected curvature Ω c o r. We see that in terms of Ω c o r this action takes the form typical for a gauge field. Finally, we give a geometrical interpretation of the corrected curvature Ω c o r. Full article
(This article belongs to the Special Issue Applications of Differential Geometry)
Open AccessArticle The Laplacian Flow of Locally Conformal Calibrated G2-Structures
Received: 8 November 2018 / Revised: 31 December 2018 / Accepted: 3 January 2019 / Published: 11 January 2019
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Abstract
We consider the Laplacian flow of locally conformal calibrated G2-structures as a natural extension to these structures of the well-known Laplacian flow of calibrated G2-structures. We study the Laplacian flow for two explicit examples of locally conformal calibrated G [...] Read more.
We consider the Laplacian flow of locally conformal calibrated G 2 -structures as a natural extension to these structures of the well-known Laplacian flow of calibrated G 2 -structures. We study the Laplacian flow for two explicit examples of locally conformal calibrated G 2 manifolds and, in both cases, we obtain a flow of locally conformal calibrated G 2 -structures, which are ancient solutions, that is they are defined on a time interval of the form ( , T ) , where T > 0 is a real number. Moreover, for each of these examples, we prove that the underlying metrics g ( t ) of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric as t goes to , and they blow-up at a finite-time singularity. Full article
(This article belongs to the Special Issue Applications of Differential Geometry)
Open AccessArticle Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV
Received: 16 November 2018 / Revised: 17 December 2018 / Accepted: 19 December 2018 / Published: 25 December 2018
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Abstract
This paper is one of a series in which we generalize our earlier results on the equivalence of existence of Calabi extremal metrics to the geodesic stability for any type I compact complex almost homogeneous manifolds of cohomogeneity one. In this paper, we [...] Read more.
This paper is one of a series in which we generalize our earlier results on the equivalence of existence of Calabi extremal metrics to the geodesic stability for any type I compact complex almost homogeneous manifolds of cohomogeneity one. In this paper, we actually carry all the earlier results to the type I cases. In Part II, we obtained a substantial amount of new Kähler–Einstein manifolds as well as Fano manifolds without Kähler–Einstein metrics. In particular, by applying Theorem 15 therein, we obtained complete results in the Theorems 3 and 4 in that paper. However, we only have partial results in Theorem 5. In this note, we provide a report of recent progress on the Fano manifolds N n , m when n > 15 and N n , m when n > 4 . We provide two pictures for these two classes of manifolds. See Theorems 1 and 2 in the last section. Moreover, we present two conjectures. Once we solve these two conjectures, the question for these two classes of manifolds will be completely solved. By applying our results to the canonical circle bundles, we also obtain Sasakian manifolds with or without Sasakian–Einstein metrics. These also provide open Calabi–Yau manifolds. Full article
(This article belongs to the Special Issue Applications of Differential Geometry)
Open AccessArticle Special Types of Locally Conformal Closed G2-Structures
Received: 30 October 2018 / Revised: 23 November 2018 / Accepted: 23 November 2018 / Published: 28 November 2018
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Abstract
Motivated by known results in locally conformal symplectic geometry, we study different classes of G2-structures defined by a locally conformal closed 3-form. In particular, we provide a complete characterization of invariant exact locally conformal closed G2-structures on simply connected [...] Read more.
Motivated by known results in locally conformal symplectic geometry, we study different classes of G 2 -structures defined by a locally conformal closed 3-form. In particular, we provide a complete characterization of invariant exact locally conformal closed G 2 -structures on simply connected Lie groups, and we present examples of compact manifolds with different types of locally conformal closed G 2 -structures. Full article
(This article belongs to the Special Issue Applications of Differential Geometry)
Open AccessArticle Exponentially Harmonic Maps into Spheres
Received: 29 October 2018 / Revised: 16 November 2018 / Accepted: 18 November 2018 / Published: 22 November 2018
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Abstract
We study smooth exponentially harmonic maps from a compact, connected, orientable Riemannian manifold M into a sphere SmRm+1. Given a codimension two totally geodesic submanifold ΣSm, we show that every nonconstant exponentially [...] Read more.
We study smooth exponentially harmonic maps from a compact, connected, orientable Riemannian manifold M into a sphere S m R m + 1 . Given a codimension two totally geodesic submanifold Σ S m , we show that every nonconstant exponentially harmonic map ϕ : M S m either meets or links Σ . If H 1 ( M , Z ) = 0 then ϕ ( M ) Σ . Full article
(This article belongs to the Special Issue Applications of Differential Geometry)
Open AccessArticle The Role of Spin(9) in Octonionic Geometry
Received: 20 September 2018 / Revised: 4 October 2018 / Accepted: 6 October 2018 / Published: 12 October 2018
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Abstract
Starting from the 2001 Thomas Friedrich’s work on Spin(9), we review some interactions between Spin(9) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin(9) canonical [...] Read more.
Starting from the 2001 Thomas Friedrich’s work on Spin ( 9 ) , we review some interactions between Spin ( 9 ) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin ( 9 ) canonical 8-form and its analogies with quaternionic geometry as well as the role of Spin ( 9 ) both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin ( 9 ) manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley–Rosenfeld planes and to three series of Grassmannians. Full article
(This article belongs to the Special Issue Applications of Differential Geometry)
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Open AccessArticle Characterizations of the Total Space (Indefinite Trans-Sasakian Manifolds) Admitting a Semi-Symmetric Metric Connection
Received: 19 August 2018 / Revised: 4 September 2018 / Accepted: 7 September 2018 / Published: 10 September 2018
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Abstract
We investigate recurrent, Lie-recurrent, and Hopf lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. In these hypersurfaces, we obtain several new results. Moreover, we characterize that the total space (an indefinite generalized Sasakian space form) with a semi-symmetric metric [...] Read more.
We investigate recurrent, Lie-recurrent, and Hopf lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. In these hypersurfaces, we obtain several new results. Moreover, we characterize that the total space (an indefinite generalized Sasakian space form) with a semi-symmetric metric connection is an indefinite Kenmotsu space form under various lightlike hypersurfaces. Full article
(This article belongs to the Special Issue Applications of Differential Geometry)

Review

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Open AccessReview Contact Semi-Riemannian Structures in CR Geometry: Some Aspects
Received: 26 September 2018 / Revised: 20 December 2018 / Accepted: 2 January 2019 / Published: 9 January 2019
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Abstract
There is one-to-one correspondence between contact semi-Riemannian structures (η,ξ,φ,g) and non-degenerate almost CR structures (H,ϑ,J). In general, a non-degenerate almost CR structure is not a CR structure, that [...] Read more.
There is one-to-one correspondence between contact semi-Riemannian structures ( η , ξ , φ , g ) and non-degenerate almost CR structures ( H , ϑ , J ) . In general, a non-degenerate almost CR structure is not a CR structure, that is, in general the integrability condition for H 1 , 0 : = X i J X , X H is not satisfied. In this paper we give a survey on some known results, with the addition of some new results, on the geometry of contact semi-Riemannian manifolds, also in the context of the geometry of Levi non-degenerate almost CR manifolds of hypersurface type, emphasizing similarities and differences with respect to the Riemannian case. Full article
(This article belongs to the Special Issue Applications of Differential Geometry)
Open AccessReview Mathematical Modeling of Rogue Waves: A Survey of Recent and Emerging Mathematical Methods and Solutions
Received: 16 May 2018 / Revised: 6 June 2018 / Accepted: 8 June 2018 / Published: 20 June 2018
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Abstract
Anomalous waves and rogue events are closely associated with irregularities and unexpected events occurring at various levels of physics, such as in optics, in oceans and in the atmosphere. Mathematical modeling of rogue waves is a highly active field of research, which has [...] Read more.
Anomalous waves and rogue events are closely associated with irregularities and unexpected events occurring at various levels of physics, such as in optics, in oceans and in the atmosphere. Mathematical modeling of rogue waves is a highly active field of research, which has evolved over the last few decades into a specialized part of mathematical physics. The applications of the mathematical models for rogue events is directly relevant to technology development for the prediction of rogue ocean waves and for signal processing in quantum units. In this survey, a comprehensive perspective of the most recent developments of methods for representing rogue waves is given, along with discussion of the devised forms and solutions. The standard nonlinear Schrödinger equation, the Hirota equation, the MMT equation and other models are discussed and their properties highlighted. This survey shows that the most recent advancement in modeling rogue waves give models that can be used to establish methods for the prediction of rogue waves in open seas, which is important for the safety and activity of marine vessels and installations. The study further puts emphasis on the difference between the methods and how the resulting models form the basis for representing rogue waves in various forms, solitary or with a wave background. This review has also a pedagogic component directed towards students and interested non-experts and forms a complete survey of the most conventional and emerging methods published until recently. Full article
(This article belongs to the Special Issue Applications of Differential Geometry)
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