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16 pages, 292 KB

Open AccessArticle

On the Classification of Totally Geodesic and Parallel Hypersurfaces of the Lie Group Nil⁴

by Guixian Huang and Jinguo Jiang

Symmetry 2025, 17(11), 1979; https://doi.org/10.3390/sym17111979 - 16 Nov 2025

Viewed by 231

This work establishes a complete algebraic classification of hypersurfaces with totally symmetric cubic form, including the Codazzi, parallel, and totally geodesic cases, on the 4-dimensional 3-step nilpotent Lie group

N i l^{4}

endowed with six left-invariant Lorentzian metrics. Combined with prior results, [...] Read more.

This work establishes a complete algebraic classification of hypersurfaces with totally symmetric cubic form, including the Codazzi, parallel, and totally geodesic cases, on the 4-dimensional 3-step nilpotent Lie group

N i l^{4}

endowed with six left-invariant Lorentzian metrics. Combined with prior results, we achieve a complete classification of such hypersurfaces on 4-dimensional nilpotent Lie groups. The core of our approach lies in the explicit derivation and solution of the Codazzi tensor equations, which directly leads to the construction of these hypersurfaces and provides their explicit parametrizations. Our main results establish the existence of Codazzi hypersurfaces on

N i l^{4}

, demonstrate the non-existence of totally geodesic hypersurfaces, specify the algebraic condition for a Codazzi hypersurface to become parallel, and provide their explicit parametrizations. This observation highlights fundamental differences between Lorentzian and Riemannian settings within hypersurface theory. This work thus clarifies the distinct geometric properties inherent to the Lorentzian cases on nilpotent Lie groups. Full article

(This article belongs to the Special Issue Symmetry in Lie Groups and Lie Algebras)

13 pages, 265 KB

Open AccessArticle

Impact of Ambient Conformal Vector Fields on Yamabe Solitons on Riemannian Hypersurfaces

by Norah Alshehri

Mathematics 2025, 13(17), 2725; https://doi.org/10.3390/math13172725 - 25 Aug 2025

Viewed by 522

Abstract

We investigate Yamabe solitons on Riemannian hypersurfaces induced by conformal vector fields in Riemannian and Lorentzian manifolds, with an emphasis on the tangential component. We show that these hypersurfaces are totally umbilical, and when the ambient manifold is Einstein, a rigidity condition emerges [...] Read more.

We investigate Yamabe solitons on Riemannian hypersurfaces induced by conformal vector fields in Riemannian and Lorentzian manifolds, with an emphasis on the tangential component. We show that these hypersurfaces are totally umbilical, and when the ambient manifold is Einstein, a rigidity condition emerges connecting the mean and scalar curvatures. Using this, we classify compact Yamabe solitons: each hypersurface is either totally geodesic or an extrinsic sphere. Additionally, we prove the non-existence of trivial Yamabe solitons on oriented hypersurfaces of higher dimension in Einstein manifolds. These results highlight the classification of compact hypersurfaces and rigidity phenomena in the ambient spaces, providing a clear understanding of the geometric structures associated with Yamabe solitons. Full article

15 pages, 295 KB

Open AccessArticle

k-Almost Newton-Conformal Ricci Solitons on Hypersurfaces Within Golden Riemannian Manifolds with Constant Golden Sectional Curvature

by Amit Kumar Rai, Majid Ali Choudhary, Mohd. Danish Siddiqi, Ghodratallah Fasihi-Ramandi, Uday Chand De and Ion Mihai

Axioms 2025, 14(8), 579; https://doi.org/10.3390/axioms14080579 - 26 Jul 2025

Viewed by 601

Abstract

The current work establishes the geometrical bearing for hypersurfaces in a Golden Riemannian manifold with constant golden sectional curvature with respect to k-almost Newton-conformal Ricci solitons. Moreover, we extensively explore the immersed r-almost Newton-conformal Ricci soliton and determine the sufficient conditions [...] Read more.

The current work establishes the geometrical bearing for hypersurfaces in a Golden Riemannian manifold with constant golden sectional curvature with respect to k-almost Newton-conformal Ricci solitons. Moreover, we extensively explore the immersed r-almost Newton-conformal Ricci soliton and determine the sufficient conditions for total geodesicity with adequate restrictions on some smooth functions using mathematical operators. Furthermore, we go over some natural conclusions in which the gradient k-almost Newton-conformal Ricci soliton on the hypersurface of the Golden Riemannian manifold becomes compact. Finally, we establish a Schur’s type inequality in terms of k-almost Newton-conformal Ricci solitons immersed in Golden Riemannian manifolds with constant golden sectional curvature. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application, 3rd Edition)

13 pages, 254 KB

Open AccessArticle

Ricci Solitons on Riemannian Hypersurfaces Generated by Torse-Forming Vector Fields in Riemannian and Lorentzian Manifolds

by Norah Alshehri and Mohammed Guediri

Axioms 2025, 14(5), 325; https://doi.org/10.3390/axioms14050325 - 23 Apr 2025

Cited by 1 | Viewed by 501

Abstract

In this paper, we examine torse-forming vector fields to characterize extrinsic spheres (that is, totally umbilical hypersurfaces with nonzero constant mean curvatures) in Riemannian and Lorentzian manifolds. First, we analyze the properties of these vector fields on Riemannian manifolds. Next, we focus on [...] Read more.

In this paper, we examine torse-forming vector fields to characterize extrinsic spheres (that is, totally umbilical hypersurfaces with nonzero constant mean curvatures) in Riemannian and Lorentzian manifolds. First, we analyze the properties of these vector fields on Riemannian manifolds. Next, we focus on Ricci solitons on Riemannian hypersurfaces induced by torse-forming vector fields of Riemannian or Lorentzian manifolds. Specifically, we show that such a hypersurface in the manifold with constant sectional curvature is either totally geodesic or an extrinsic sphere. Full article

14 pages, 305 KB

Open AccessArticle

Rigidity and Triviality of Gradient r-Almost Newton-Ricci-Yamabe Solitons

by Mohd Danish Siddiqi and Fatemah Mofarreh

Mathematics 2024, 12(20), 3173; https://doi.org/10.3390/math12203173 - 10 Oct 2024

Viewed by 1026

Abstract

In this paper, we develop the concept of gradient r-Almost Newton-Ricci-Yamabe solitons (in brief, gradient r-ANRY solitons) immersed in a Riemannian manifold. We deduce the minimal and totally geodesic criteria for the hypersurface of a Riemannian manifold in terms of the [...] Read more.

In this paper, we develop the concept of gradient r-Almost Newton-Ricci-Yamabe solitons (in brief, gradient r-ANRY solitons) immersed in a Riemannian manifold. We deduce the minimal and totally geodesic criteria for the hypersurface of a Riemannian manifold in terms of the gradient r-ANRY soliton. We also exhibit a Schur-type inequality and discuss the triviality of the gradient r-ANRY soliton in the case of a compact manifold. Finally, we demonstrate the completeness and noncompactness of the r-Newton-Ricci-Yamabe soliton on the hypersurface of the Riemannian manifold. Full article

(This article belongs to the Special Issue Differentiable Manifolds and Geometric Structures)

20 pages, 316 KB

Open AccessArticle

On Geodesic Triangles in Non-Euclidean Geometry

by Antonella Nannicini and Donato Pertici

Foundations 2024, 4(4), 468-487; https://doi.org/10.3390/foundations4040030 - 26 Sep 2024

Viewed by 1963

Abstract

In this paper, we study centroids, orthocenters, circumcenters, and incenters of geodesic triangles in non-Euclidean geometry, and we discuss the existence of the Euler line in this context. Moreover, we give simple proofs of the existence of a totally geodesic 2-dimensional submanifold containing [...] Read more.

In this paper, we study centroids, orthocenters, circumcenters, and incenters of geodesic triangles in non-Euclidean geometry, and we discuss the existence of the Euler line in this context. Moreover, we give simple proofs of the existence of a totally geodesic 2-dimensional submanifold containing a given geodesic triangle in the hyperbolic or spherical 3-dimensional geometry. Full article

(This article belongs to the Section Mathematical Sciences)

8 pages, 221 KB

Open AccessArticle

The Shape Operator of Real Hypersurfaces in S⁶(1)

by Djordje Kocić and Miroslava Antić

Mathematics 2024, 12(11), 1668; https://doi.org/10.3390/math12111668 - 27 May 2024

Viewed by 1096

Abstract

The aim of the paper is to present two results concerning real hypersurfaces in the six-dimensional sphere

S^{6} (1)

. More precisely, we prove that real hypersurfaces with the Lie-parallel shape operator A must be totally geodesic hyperspheres. Additionally, we [...] Read more.

The aim of the paper is to present two results concerning real hypersurfaces in the six-dimensional sphere

S^{6} (1)

. More precisely, we prove that real hypersurfaces with the Lie-parallel shape operator A must be totally geodesic hyperspheres. Additionally, we classify real hypersurfaces in a nearly Kähler sphere

S^{6} (1)

whose Lie derivative of the shape operator coincides with its covariant derivative. Full article

(This article belongs to the Special Issue Differential Geometry, Geometric Analysis and Their Related Applications)

20 pages, 380 KB

Open AccessReview

Integral Formulas for Almost Product Manifolds and Foliations

by Vladimir Rovenski

Mathematics 2022, 10(19), 3645; https://doi.org/10.3390/math10193645 - 5 Oct 2022

Cited by 2 | Viewed by 1891

Abstract

Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying [...] Read more.

Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying such problems as (i) the existence and characterization of foliations with a given geometric property, such as being totally geodesic, minimal or totally umbilical; (ii) prescribing the generalized mean curvatures of the leaves of a foliation; (iii) minimizing volume-like functionals defined for tensors on foliated manifolds. We start from the series of integral formulas for codimension one foliations of Riemannian and metric-affine manifolds, and then we consider integral formulas for regular and singular foliations of arbitrary codimension. In the second part of the article, we represent integral formulas with the mixed scalar curvature of an almost multi-product structure on Riemannian and metric-affine manifolds, give applications to hypersurfaces of space forms with

k = 2, 3

distinct principal curvatures of constant multiplicities and then discuss integral formulas for foliations or distributions on sub-Riemannian manifolds. Full article

(This article belongs to the Section E: Applied Mathematics)

9 pages, 269 KB

Open AccessArticle

On Minimal Hypersurfaces of a Unit Sphere

by Amira Ishan, Sharief Deshmukh, Ibrahim Al-Dayel and Cihan Özgür

Mathematics 2021, 9(24), 3161; https://doi.org/10.3390/math9243161 - 8 Dec 2021

Cited by 1 | Viewed by 2885

Abstract

Minimal compact hypersurface in the unit sphere

S^{n + 1}

having squared length of shape operator

{∥A∥}^{2} < n

are totally geodesic and with

{∥A∥}^{2} = n

are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance [...] Read more.

Minimal compact hypersurface in the unit sphere

S^{n + 1}

having squared length of shape operator

{∥A∥}^{2} < n

are totally geodesic and with

{∥A∥}^{2} = n

are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in

S^{n + 1}

. One finds a naturally induced vector field w called the associated vector field and a smooth function

ρ

called support function on the hypersurface M of

S^{n + 1}

. It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in

S^{5}

to be totally geodesic is that the support function

ρ

is a non-trivial solution of static perfect fluid equation. Additionally, this result holds for minimal compact hypersurfaces in

S^{n + 1}

, (

n > 2

), provided the scalar curvature

τ

is a constant on integral curves of w. Yet other classification of totally geodesic hypersurfaces among minimal compact hypersurfaces in

S^{n + 1}

is obtained using the associated vector field w an eigenvector of rough Laplace operator. Finally, a characterization of Clifford hypersurfaces is found using an upper bound on the integral of Ricci curvature in the direction of the vector field

A w

. Full article

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

35 pages, 443 KB

Open AccessReview

The Rigging Technique for Null Hypersurfaces

by Manuel Gutiérrez and Benjamín Olea

Axioms 2021, 10(4), 284; https://doi.org/10.3390/axioms10040284 - 29 Oct 2021

Cited by 8 | Viewed by 2671

Abstract

Starting from the main definitions, we review the rigging technique for null hypersurfaces theory and most of its main properties. We make some applications to illustrate it. On the one hand, we show how we can use it to show properties of null [...] Read more.

Starting from the main definitions, we review the rigging technique for null hypersurfaces theory and most of its main properties. We make some applications to illustrate it. On the one hand, we show how we can use it to show properties of null hypersurfaces, with emphasis in null cones, totally geodesic, totally umbilic, and compact null hypersurfaces. On the other hand, we show the interplay with the ambient space, including its influence in causality theory. Full article

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

15 pages, 308 KB

Open AccessArticle

On the Canonical Foliation of an Indefinite Locally Conformal Kähler Manifold with a Parallel Lee Form

by Elisabetta Barletta, Sorin Dragomir and Francesco Esposito

Mathematics 2021, 9(4), 333; https://doi.org/10.3390/math9040333 - 7 Feb 2021

Cited by 1 | Viewed by 1981

Abstract

We study the semi-Riemannian geometry of the foliation

F

of an indefinite locally conformal Kähler (l.c.K.) manifold M, given by the Pfaffian equation

ω = 0

, provided that

\nabla ω = 0

and

c = ∥ ω ∥ \neq 0

( [...] Read more.

We study the semi-Riemannian geometry of the foliation

F

of an indefinite locally conformal Kähler (l.c.K.) manifold M, given by the Pfaffian equation

ω = 0

, provided that

\nabla ω = 0

and

c = ∥ ω ∥ \neq 0

(

ω

is the Lee form of M). If M is conformally flat then every leaf of

F

is shown to be a totally geodesic semi-Riemannian hypersurface in M, and a semi-Riemannian space form of sectional curvature

c / 4

, carrying an indefinite c-Sasakian structure. As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem any geodesically complete, conformally flat, indefinite Vaisman manifold of index

2 s

,

0 < s < n

, is locally biholomorphically homothetic to an indefinite complex Hopf manifold

C H_{s}^{n} (λ)

,

0 < λ < 1

, equipped with the indefinite Boothby metric

g_{s, n}

. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

16 pages, 298 KB

Open AccessArticle

Concurrent Vector Fields on Lightlike Hypersurfaces

by Erol Kılıç, Mehmet Gülbahar and Ecem Kavuk

Mathematics 2021, 9(1), 59; https://doi.org/10.3390/math9010059 - 29 Dec 2020

Cited by 10 | Viewed by 2147

Abstract

Concurrent vector fields lying on lightlike hypersurfaces of a Lorentzian manifold are investigated. Obtained results dealing with concurrent vector fields are discussed for totally umbilical lightlike hypersurfaces and totally geodesic lightlike hypersurfaces. Furthermore, Ricci soliton lightlike hypersurfaces admitting concurrent vector fields are studied [...] Read more.

Concurrent vector fields lying on lightlike hypersurfaces of a Lorentzian manifold are investigated. Obtained results dealing with concurrent vector fields are discussed for totally umbilical lightlike hypersurfaces and totally geodesic lightlike hypersurfaces. Furthermore, Ricci soliton lightlike hypersurfaces admitting concurrent vector fields are studied and some characterizations for this frame of hypersurfaces are obtained. Full article

(This article belongs to the Section B: Geometry and Topology)

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