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21 pages, 331 KiB

Open AccessArticle

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 249

Abstract

In this paper, we investigate the sectional contravariant curvature of a doubly warped product manifold

(_{f} B \times_{b} F, \tilde{g}, Π = Π_{B} + Π_{F})

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

(_{f} B \times_{b} F, \tilde{g}, Π = Π_{B} + Π_{F})

equipped with a product Poisson structure

Π

, using warping functions and sectional curvatures of its factor manifolds

(B, {\tilde{g}}_{B}, Π_{B})

and

(F, {\tilde{g}}_{F}, Π_{F})

. Qualar and null sectional contravariant curvatures of

(_{f} B \times_{b} F, \tilde{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

Open AccessArticle

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 386

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

R^{3}

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

R^{3}

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

Open AccessFeature PaperArticle

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 333

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

Open AccessArticle

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 - 06 Feb 2024

Viewed by 509

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

Open AccessArticle

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 7 | Viewed by 760

Abstract

The twisted hypersurfaces

x

with the

(0, 0, 0, 0, 1)

rotating axis in five-dimensional Euclidean space

E^{5}

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

E^{5}

is considered. The fundamental forms, the Gauss map, and the shape operator of

x

are calculated. In

E^{5}

, 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

Open AccessArticle

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 587

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

Open AccessArticle

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 717

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

Open AccessArticle

A Hypersurfaces of Revolution Family in the Five-Dimensional Pseudo-Euclidean Space

E_{2}^{5}

by Yanlin Li and Erhan Güler

Mathematics 2023, 11(15), 3427; https://doi.org/10.3390/math11153427 - 07 Aug 2023

Cited by 19 | Viewed by 1290

Abstract

We present a family of hypersurfaces of revolution distinguished by four parameters in the five-dimensional pseudo-Euclidean space

E_{2}^{5}

. 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

E_{2}^{5}

. 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 \times 5

matrix. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications)

17 pages, 328 KiB

Open AccessArticle

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 590

Abstract

In this article, we classify the developable surfaces in three-dimensional Euclidean space

R^{3}

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

R^{3}

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

Open AccessArticle

CDE’ Inequality on Graphs with Unbounded Laplacian

by Desheng Hong and Chao Gong

Mathematics 2023, 11(9), 2138; https://doi.org/10.3390/math11092138 - 03 May 2023

Viewed by 689

Abstract

In this paper, we derive the gradient estimates of semigroups in terms of the modified curvature-dimension inequality

C D E^{'}

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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