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10 pages, 302 KiB

Open AccessFeature PaperArticle

On the Geometry in the Large of Einstein-like Manifolds

by Josef Mikeš, Lenka Rýparová, Sergey Stepanov and Irina Tsyganok

Mathematics 2022, 10(13), 2208; https://doi.org/10.3390/math10132208 - 24 Jun 2022

Viewed by 1621

Abstract

Gray has presented the invariant orthogonal irreducible decomposition of the space of all covariant tensors of rank 3, obeying only the identities of the gradient of the Ricci tensor. This decomposition introduced the seven classes of Einstein-like manifolds, the Ricci tensors of which [...] Read more.

Gray has presented the invariant orthogonal irreducible decomposition of the space of all covariant tensors of rank 3, obeying only the identities of the gradient of the Ricci tensor. This decomposition introduced the seven classes of Einstein-like manifolds, the Ricci tensors of which fulfill the defining condition of each subspace. The large-scale geometry of such manifolds has been studied by many geometers using the classical Bochner technique. However, the scope of this method is limited to compact Riemannian manifolds. In the present paper, we prove several Liouville-type theorems for certain classes of Einstein-like complete manifolds. This represents an illustration of the new possibilities of geometric analysis. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Special Structures)

12 pages, 292 KiB

Open AccessArticle

Geodesic Mappings onto Generalized m-Ricci-Symmetric Spaces

by Volodymyr Berezovski, Yevhen Cherevko, Irena Hinterleitner and Patrik Peška

Mathematics 2022, 10(13), 2165; https://doi.org/10.3390/math10132165 - 21 Jun 2022

Cited by 2 | Viewed by 1298

Abstract

In this paper, we study geodesic mappings of spaces with affine connections onto generalized 2-, 3-, and m-Ricci-symmetric spaces. In either case, the main equations for the mappings are obtained as a closed system of linear differential equations of the Cauchy type [...] Read more.

In this paper, we study geodesic mappings of spaces with affine connections onto generalized 2-, 3-, and m-Ricci-symmetric spaces. In either case, the main equations for the mappings are obtained as a closed system of linear differential equations of the Cauchy type in the covariant derivatives. For the systems, we have found the maximum number of essential parameters on which the solutions depend. These results generalize the properties of geodesic mappings onto symmetric, recurrent, and also 2-, 3-, and m-(Ricci-)symmetric spaces with affine connections. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Special Structures)

11 pages, 294 KiB

Open AccessArticle

A Note on Generalized Quasi-Einstein and (λ, n + m)-Einstein Manifolds with Harmonic Conformal Tensor

by Sameh Shenawy, Carlo Alberto Mantica, Luca Guido Molinari and Nasser Bin Turki

Mathematics 2022, 10(10), 1731; https://doi.org/10.3390/math10101731 - 18 May 2022

Cited by 2 | Viewed by 1643

Abstract

Sufficient conditions for a Lorentzian generalized quasi-Einstein manifold

(M, g, f, μ)

to be a generalized Robertson–Walker spacetime with Einstein fibers are derived. The Ricci tensor in this case gains the perfect fluid form. Likewise, it is proven that a [...] Read more.

Sufficient conditions for a Lorentzian generalized quasi-Einstein manifold

(M, g, f, μ)

to be a generalized Robertson–Walker spacetime with Einstein fibers are derived. The Ricci tensor in this case gains the perfect fluid form. Likewise, it is proven that a

(λ, n + m)

-Einstein manifold

(M, g, w)

having harmonic Weyl tensor,

(\nabla^{j} w) (\nabla^{m} w) C_{j k l m} = 0

and

\nabla_{l} w \nabla^{l} w < 0

reduces to a perfect fluid generalized Robertson–Walker spacetime with Einstein fibers. Finally,

(M, g, w)

reduces to a perfect fluid manifold if

φ = - m \nabla (\ln w)

is a

φ (R i c)

-vector field on M and to an Einstein manifold if

ψ = \nabla w

is a

ψ (R i c)

-vector field on M. Some consequences of these results are considered. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Special Structures)

11 pages, 262 KiB

Open AccessArticle

Geodesic Mappings of Semi-Riemannian Manifolds with a Degenerate Metric

by Igor G. Shandra and Josef Mikeš

Mathematics 2022, 10(1), 154; https://doi.org/10.3390/math10010154 - 5 Jan 2022

Viewed by 1688

Abstract

This article introduces the concept of geodesic mappings of manifolds with idempotent pseudo-connections. The basic equations of canonical geodesic mappings of manifolds with completely idempotent pseudo-connectivity and semi-Riemannian manifolds with a degenerate metric are obtained. It is proved that semi-Riemannian manifolds admitting concircular [...] Read more.

This article introduces the concept of geodesic mappings of manifolds with idempotent pseudo-connections. The basic equations of canonical geodesic mappings of manifolds with completely idempotent pseudo-connectivity and semi-Riemannian manifolds with a degenerate metric are obtained. It is proved that semi-Riemannian manifolds admitting concircular fields admit completely canonical geodesic mappings and form a closed class with respect to these mappings. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Special Structures)

10 pages, 637 KiB

Open AccessArticle

Nullcone Fronts of Spacelike Framed Curves in Minkowski 3-Space

by Pengcheng Li and Donghe Pei

Mathematics 2021, 9(22), 2939; https://doi.org/10.3390/math9222939 - 18 Nov 2021

Cited by 9 | Viewed by 2161

Abstract

The investigation of objects in Minkowski space is of great significance, especially for those objects with mathematical and physical backgrounds. In this paper, we study nullcone fronts, which are formed by the light rays emitted from points on a spacelike curve. However, if [...] Read more.

The investigation of objects in Minkowski space is of great significance, especially for those objects with mathematical and physical backgrounds. In this paper, we study nullcone fronts, which are formed by the light rays emitted from points on a spacelike curve. However, if the spacelike curve is singular, then we cannot use the usual tools and methods to study related issues. To solve these problems, we show the definition of spacelike framed curves in Minkowski 3-space, whose original curves may contain singularities. Then, the singularities of the nullcone fronts are characterized by using framed curvatures of spacelike framed curves. Finally, we exhibit some examples to illustrate our results. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Special Structures)

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7 pages, 623 KiB

Open AccessArticle

A Novel Analysis of the Smooth Curve with Constant Width Based on a Time Delay System

by Teng Fu and Yusheng Zhou

Mathematics 2021, 9(10), 1131; https://doi.org/10.3390/math9101131 - 17 May 2021

Cited by 2 | Viewed by 2234

Abstract

In this paper, we analyze the

C^{\infty}

smooth curve of constant width using the characteristic equation of a time delay system. We prove that a closed convex curve must be a circle if it is still a smooth curve of constant width [...] Read more.

In this paper, we analyze the

C^{\infty}

smooth curve of constant width using the characteristic equation of a time delay system. We prove that a closed convex curve must be a circle if it is still a smooth curve of constant width after taking any number of derivatives. Finally, the simulation results are presented for analyzing the influence of derivative orders on a smooth non-circular curve of constant width. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Special Structures)

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9 pages, 246 KiB

Open AccessArticle

Generalized Affine Connections Associated with the Space of Centered Planes

by Olga Belova

Mathematics 2021, 9(7), 782; https://doi.org/10.3390/math9070782 - 5 Apr 2021

Cited by 4 | Viewed by 1964

Abstract

Our purpose is to study a space

Π

of centered m-planes in n-projective space. Generalized fiberings (with semi-gluing) are investigated. Planar and normal affine connections associated with the space

Π

are set in the generalized fiberings. Fields of these affine connection [...] Read more.

Our purpose is to study a space

Π

of centered m-planes in n-projective space. Generalized fiberings (with semi-gluing) are investigated. Planar and normal affine connections associated with the space

Π

are set in the generalized fiberings. Fields of these affine connection objects define torsion and curvature tensors. The canonical cases of planar and normal generalized affine connections are considered. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Special Structures)

Review

Jump to: Research

22 pages, 343 KiB

Open AccessReview

Types of Submanifolds in Metallic Riemannian Manifolds: A Short Survey

by Cristina E. Hretcanu and Adara M. Blaga

Mathematics 2021, 9(19), 2467; https://doi.org/10.3390/math9192467 - 3 Oct 2021

Cited by 4 | Viewed by 2241

Abstract

We provide a brief survey on the properties of submanifolds in metallic Riemannian manifolds. We focus on slant, semi-slant and hemi-slant submanifolds in metallic Riemannian manifolds and, in particular, on invariant, anti-invariant and semi-invariant submanifolds. We also describe the warped product bi-slant and, [...] Read more.

We provide a brief survey on the properties of submanifolds in metallic Riemannian manifolds. We focus on slant, semi-slant and hemi-slant submanifolds in metallic Riemannian manifolds and, in particular, on invariant, anti-invariant and semi-invariant submanifolds. We also describe the warped product bi-slant and, in particular, warped product semi-slant and warped product hemi-slant submanifolds in locally metallic Riemannian manifolds, obtaining some results regarding the existence and nonexistence of non-trivial semi-invariant, semi-slant and hemi-slant warped product submanifolds. We illustrate all these by suitable examples. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Special Structures)

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