Research

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9 pages, 250 KiB

Open AccessArticle

General Relativity with a Positive Cosmological Constant Λ as a Gauge Theory

by Marta Dudek and Janusz Garecki

Axioms 2019, 8(1), 24; https://doi.org/10.3390/axioms8010024 - 21 Feb 2019

Viewed by 2566

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)

15 pages, 280 KiB

Open AccessArticle

The Laplacian Flow of Locally Conformal Calibrated G₂-Structures

by Marisa Fernández, Victor Manero and Jonatan Sánchez

Axioms 2019, 8(1), 7; https://doi.org/10.3390/axioms8010007 - 11 Jan 2019

Cited by 1 | Viewed by 2352

Abstract

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 [...] 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

(- \infty, 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

- \infty

, and they blow-up at a finite-time singularity. Full article

(This article belongs to the Special Issue Applications of Differential Geometry)

9 pages, 257 KiB

Open AccessArticle

Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

by Zhuang-Dan Daniel Guan, Pilar Orellana and Anthony Van

Axioms 2019, 8(1), 2; https://doi.org/10.3390/axioms8010002 - 25 Dec 2018

Viewed by 3592

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)

14 pages, 328 KiB

Open AccessArticle

Special Types of Locally Conformal Closed G₂-Structures

by Giovanni Bazzoni and Alberto Raffero

Axioms 2018, 7(4), 90; https://doi.org/10.3390/axioms7040090 - 28 Nov 2018

Cited by 1 | Viewed by 2717

Abstract

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 [...] 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)

7 pages, 311 KiB

Open AccessArticle

Exponentially Harmonic Maps into Spheres

by Sorin Dragomir and Francesco Esposito

Axioms 2018, 7(4), 88; https://doi.org/10.3390/axioms7040088 - 22 Nov 2018

Cited by 1 | Viewed by 2218

Abstract

We study smooth exponentially harmonic maps from a compact, connected, orientable Riemannian manifold M into a sphere

S^{m} \subset R^{m + 1}

. Given a codimension two totally geodesic submanifold

Σ \subset S^{m}

, 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} \subset R^{m + 1}

. Given a codimension two totally geodesic submanifold

Σ \subset S^{m}

, we show that every nonconstant exponentially harmonic map

ϕ : M \to S^{m}

either meets or links

Σ

. If

H^{1} (M, Z) = 0

then

ϕ (M) \cap Σ \neq \emptyset

. Full article

(This article belongs to the Special Issue Applications of Differential Geometry)

32 pages, 461 KiB

Open AccessArticle

The Role of Spin(9) in Octonionic Geometry

by Maurizio Parton and Paolo Piccinni

Axioms 2018, 7(4), 72; https://doi.org/10.3390/axioms7040072 - 12 Oct 2018

Cited by 3 | Viewed by 3280

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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15 pages, 232 KiB

Open AccessArticle

Characterizations of the Total Space (Indefinite Trans-Sasakian Manifolds) Admitting a Semi-Symmetric Metric Connection

by Dae Ho Jin and Jae Won Lee

Axioms 2018, 7(3), 68; https://doi.org/10.3390/axioms7030068 - 10 Sep 2018

Viewed by 2851

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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50 pages, 598 KiB

Open AccessEditor’s ChoiceReview

Contact Semi-Riemannian Structures in CR Geometry: Some Aspects

by Domenico Perrone

Axioms 2019, 8(1), 6; https://doi.org/10.3390/axioms8010006 - 09 Jan 2019

Cited by 10 | Viewed by 3338

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 \in 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)

19 pages, 2403 KiB

Open AccessReview

Mathematical Modeling of Rogue Waves: A Survey of Recent and Emerging Mathematical Methods and Solutions

by Sergio Manzetti

Axioms 2018, 7(2), 42; https://doi.org/10.3390/axioms7020042 - 20 Jun 2018

Cited by 11 | Viewed by 5635

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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