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14 pages, 3046 KB

Open AccessArticle

Parallel Hypersurfaces in 𝔼⁴ and Their Applications to Rotational Hypersurfaces

by Sezgin Büyükkütük, Ilim Kişi, Günay Öztürk and Emre Kişi

Mathematics 2025, 13(22), 3684; https://doi.org/10.3390/math13223684 - 17 Nov 2025

Viewed by 354

This study explores parallel hypersurfaces in four-dimensional Euclidean space

E^{4}

, deriving explicit expressions for their Gaussian and mean curvatures in terms of the curvature functions of the base hypersurface. We identify conditions under which these parallel hypersurfaces are flat or minimal. [...] Read more.

This study explores parallel hypersurfaces in four-dimensional Euclidean space

E^{4}

, deriving explicit expressions for their Gaussian and mean curvatures in terms of the curvature functions of the base hypersurface. We identify conditions under which these parallel hypersurfaces are flat or minimal. The theory is applied to several key hypersurfaces, including rotational hypersurfaces, hyperspheres, catenoidal hypersurfaces, and helicoidal hypersurfaces, with detailed curvature computations and visualizations. These results not only extend classical curvature relations into higher-dimensional spaces but also offer valuable insights into curvature transformations, with practical applications in both theoretical and computational geometry. Full article

(This article belongs to the Special Issue Differential Geometry, Geometric Analysis and Their Related Applications, 2nd Edition)

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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

Abstract

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)

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)

10 pages, 282 KB

Open AccessArticle

Weak Nearly Sasakian and Weak Nearly Cosymplectic Manifolds

by Vladimir Rovenski

Mathematics 2023, 11(20), 4377; https://doi.org/10.3390/math11204377 - 21 Oct 2023

Cited by 7 | Viewed by 1843

Abstract

Weak contact metric structures on a smooth manifold, introduced by V. Rovenski and R. Wolak in 2022, have provided new insight into the theory of classical structures. In this paper, we define new structures of this kind (called weak nearly Sasakian and weak [...] Read more.

Weak contact metric structures on a smooth manifold, introduced by V. Rovenski and R. Wolak in 2022, have provided new insight into the theory of classical structures. In this paper, we define new structures of this kind (called weak nearly Sasakian and weak nearly cosymplectic and nearly Kähler structures), study their geometry and give applications to Killing vector fields. We introduce weak nearly Kähler manifolds (generalizing nearly Kähler manifolds), characterize weak nearly Sasakian and weak nearly cosymplectic hypersurfaces in such Riemannian manifolds and prove that a weak nearly cosymplectic manifold with parallel Reeb vector field is locally the Riemannian product of a real line and a weak nearly Kähler manifold. Full article

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

12 pages, 276 KB

Open AccessFeature PaperArticle

Results of Hyperbolic Ricci Solitons

by Adara M. Blaga and Cihan Özgür

Symmetry 2023, 15(8), 1548; https://doi.org/10.3390/sym15081548 - 6 Aug 2023

Cited by 10 | Viewed by 2115

Abstract

We obtain some properties of a hyperbolic Ricci soliton with certain types of potential vector fields, and we point out some conditions when it reduces to a trivial Ricci soliton. We also study those soliton submanifolds whose vector fields are the tangential components [...] Read more.

We obtain some properties of a hyperbolic Ricci soliton with certain types of potential vector fields, and we point out some conditions when it reduces to a trivial Ricci soliton. We also study those soliton submanifolds whose vector fields are the tangential components of a concurrent vector field on the ambient manifold, and in particular, we show that a totally umbilical hyperbolic Ricci soliton is an Einstein manifold. We prove that if the hyperbolic Ricci soliton hypersurface of a Riemannian manifold of constant curvature and endowed with a concurrent vector field has a parallel shape operator, then it is a metallic-shaped hypersurface, and we determine some conditions for it to be minimal. Moreover, we show that it is also a pseudosymmetric hypersurface. Full article

14 pages, 285 KB

Open AccessArticle

The ∗-Ricci Operator on Hopf Real Hypersurfaces in the Complex Quadric

by Rongsheng Ma and Donghe Pei

Mathematics 2023, 11(1), 90; https://doi.org/10.3390/math11010090 - 26 Dec 2022

Cited by 1 | Viewed by 1640

Abstract

We study the ∗-Ricci operator on Hopf real hypersurfaces in the complex quadric. We prove that for Hopf real hypersurfaces in the complex quadric, the ∗-Ricci tensor is symmetric if and only if the unit normal vector field is singular. In the following, [...] Read more.

We study the ∗-Ricci operator on Hopf real hypersurfaces in the complex quadric. We prove that for Hopf real hypersurfaces in the complex quadric, the ∗-Ricci tensor is symmetric if and only if the unit normal vector field is singular. In the following, we obtain that if the ∗-Ricci tensor of Hopf real hypersurfaces in the complex quadric is symmetric, then the ∗-Ricci operator is both Reeb-flow-invariant and Reeb-parallel. As the correspondence to the semi-symmetric Ricci tensor, we give a classification of real hypersurfaces in the complex quadric with the semi-symmetric ∗-Ricci tensor. Full article

(This article belongs to the Special Issue Submanifolds in Metric Manifolds)

12 pages, 286 KB

Open AccessArticle

Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in S⁶(1)

by Miroslava Antić and Djordje Kocić

Mathematics 2022, 10(13), 2271; https://doi.org/10.3390/math10132271 - 29 Jun 2022

Cited by 16 | Viewed by 2094

Abstract

It is well known that the sphere

S^{6} (1)

admits an almost complex structure J which is nearly Kähler. If M is a hypersurface of an almost Hermitian manifold with a unit normal vector field N, the tangent vector [...] Read more.

It is well known that the sphere

S^{6} (1)

admits an almost complex structure J which is nearly Kähler. If M is a hypersurface of an almost Hermitian manifold with a unit normal vector field N, the tangent vector field

ξ = - J N

is said to be characteristic or the Reeb vector field. The Jacobi operator with respect to

ξ

is called the structure Jacobi operator, and is denoted by

l = R (\cdot, ξ) ξ

, where R is the curvature tensor on M. The study of Riemannian submanifolds in different ambient spaces by means of their Jacobi operators has been highly active in recent years. In particular, many recent results deal with questions around the existence of hypersurfaces with a structure Jacobi operator that satisfies conditions related to their parallelism. In the present paper, we study the parallelism of the structure Jacobi operator of real hypersurfaces in the nearly Kähler sphere

S^{6} (1)

. More precisely, we prove that such real hypersurfaces do not exist. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds: Volume II)

7 pages, 258 KB

Open AccessFeature PaperArticle

Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector

by Bang-Yen Chen

Mathematics 2019, 7(8), 710; https://doi.org/10.3390/math7080710 - 6 Aug 2019

Cited by 11 | Viewed by 3569

Abstract

The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in [...] Read more.

The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in

E^{3}

. Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in

E^{4}

; and Dimitric proved that biharmonic hypersurfaces in

E^{m}

with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for

δ (2)

-ideal and

δ (3)

-ideal hypersurfaces in

E^{m}

. Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in

E^{m}

with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors. Full article

(This article belongs to the Special Issue Complex and Contact Manifolds)

18 pages, 329 KB

Open AccessFeature PaperArticle

Solutions Modulo p of Gauss–Manin Differential Equations for Multidimensional Hypergeometric Integrals and Associated Bethe Ansatz

by Alexander Varchenko

Mathematics 2017, 5(4), 52; https://doi.org/10.3390/math5040052 - 17 Oct 2017

Cited by 10 | Viewed by 3082

Abstract

We consider the Gauss–Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallel to themselves. We reduce these equations modulo a prime integer p and construct polynomial solutions of the new differential equations as p-analogs of [...] Read more.

We consider the Gauss–Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallel to themselves. We reduce these equations modulo a prime integer p and construct polynomial solutions of the new differential equations as p-analogs of the initial hypergeometric integrals. In some cases, we interpret the p-analogs of the hypergeometric integrals as sums over points of hypersurfaces defined over the finite field Fp. This interpretation is similar to the classical interpretation by Yu. I. Manin of the number of points on an elliptic curve depending on a parameter as a solution of a Gauss hypergeometric differential equation. We discuss the associated Bethe ansatz. Full article

12 pages, 110 KB

Open AccessArticle

Time-Symmetric Boundary Conditions and Quantum Foundations

by Ken Wharton

Symmetry 2010, 2(1), 272-283; https://doi.org/10.3390/sym2010272 - 8 Mar 2010

Cited by 17 | Viewed by 8048

Abstract

Despite the widely-held premise that initial boundary conditions (BCs) corresponding to measurements/interactions can fully specify a physical subsystem, a literal reading of Hamilton’s principle would imply that both initial and final BCs are required (or more generally, a BC on a closed hypersurface [...] Read more.

Despite the widely-held premise that initial boundary conditions (BCs) corresponding to measurements/interactions can fully specify a physical subsystem, a literal reading of Hamilton’s principle would imply that both initial and final BCs are required (or more generally, a BC on a closed hypersurface in spacetime). Such a time-symmetric perspective of BCs, as applied to classical fields, leads to interesting parallels with quantum theory. This paper will map out some of the consequences of this counter-intuitive premise, as applied to covariant classical fields. The most notable result is the contextuality of fields constrained in this manner, naturally bypassing the usual arguments against so-called “realistic” interpretations of quantum phenomena. Full article

(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)

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