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15 pages, 295 KiB

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 241

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, 1294 KiB

Open AccessArticle

From Complex to Quaternions: Proof of the Riemann Hypothesis and Applications to Bose–Einstein Condensates

by Jau Tang

Symmetry 2025, 17(7), 1134; https://doi.org/10.3390/sym17071134 - 15 Jul 2025

Viewed by 582

Abstract

We present novel proofs of the Riemann hypothesis by extending the standard complex Riemann zeta function into a quaternionic algebraic framework. Utilizing λ-regularization, we construct a symmetrized form that ensures analytic continuation and restores critical-line reflection symmetry, a key structural property of the [...] Read more.

We present novel proofs of the Riemann hypothesis by extending the standard complex Riemann zeta function into a quaternionic algebraic framework. Utilizing λ-regularization, we construct a symmetrized form that ensures analytic continuation and restores critical-line reflection symmetry, a key structural property of the Riemann ξ(s) function. This formulation reveals that all nontrivial zeros of the zeta function must lie along the critical line Re(s) = 1/2, offering a constructive and algebraic resolution to this fundamental conjecture. Our method is built on convexity and symmetrical principles that generalize naturally to higher-dimensional hypercomplex spaces. We also explore the broader implications of this framework in quantum statistical physics. In particular, the λ-regularized quaternionic zeta function governs thermodynamic properties and phase transitions in Bose–Einstein condensates. This quaternionic extension of the zeta function encodes oscillatory behavior and introduces critical hypersurfaces that serve as higher-dimensional analogues of the classical critical line. By linking the spectral features of the zeta function to measurable physical phenomena, our work uncovers a profound connection between analytic number theory, hypercomplex geometry, and quantum field theory, suggesting a unified structure underlying prime distributions and quantum coherence. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials: 3rd Edition)

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16 pages, 253 KiB

Open AccessArticle

\tilde{J}

-Tangent Affine Hypersurfaces with an Induced Almost Paracontact Structure

by Zuzanna Szancer

Symmetry 2025, 17(6), 806; https://doi.org/10.3390/sym17060806 - 22 May 2025

Viewed by 322

Abstract

The subjects of our study are affine hypersurfaces

f : M \to R^{2 n + 2}

considered with a transversal vector field C, which is

\tilde{J}

-tangent. By

\tilde{J}

we understand the canonical paracomplex structure on [...] Read more.

The subjects of our study are affine hypersurfaces

f : M \to R^{2 n + 2}

considered with a transversal vector field C, which is

\tilde{J}

-tangent. By

\tilde{J}

we understand the canonical paracomplex structure on

R^{2 n + 2}

. The vector field C induces on the hypersurface f an almost paracontact structure

(φ, ξ, η)

. We obtain a complete classification of hypersurfaces admitting a metric induced almost paracontact structure with respect to the second fundamental form. We show that, in this case, the

\tilde{J}

-tangent transversal vector field is restricted to centroaffine and the hypersurface must be a piece of hyperquadric. It is demonstrated that these hyperquadrics have a very specific form. A three-dimensional example is also given. Moreover, we establish an equivalence relation between almost paracontact metric structures, para

α

-contact metric structures, and para

α

-Sasakian structures. Methods of affine differential geometry, as well as paracomplex/paracontact geometry, are used. Full article

(This article belongs to the Section Mathematics)

15 pages, 280 KiB

Open AccessArticle

Integral Formulae and Applications for Compact Riemannian Hypersurfaces in Riemannian and Lorentzian Manifolds Admitting Concircular Vector Fields

by Mona Bin-Asfour, Kholoud Saad Albalawi and Mohammed Guediri

Mathematics 2025, 13(10), 1672; https://doi.org/10.3390/math13101672 - 20 May 2025

Viewed by 400

Abstract

This paper investigates compact Riemannian hypersurfaces immersed in

(n + 1)

-dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal vector fields (CCVFs). We focus on the support function of the hypersurface, which is defined [...] Read more.

This paper investigates compact Riemannian hypersurfaces immersed in

(n + 1)

-dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal vector fields (CCVFs). We focus on the support function of the hypersurface, which is defined as the component of the conformal vector field along the unit-normal vector field, and derive an expression for its Laplacian. Using this, we establish integral formulae for hypersurfaces admitting CCVFs. These results are then extended to compact Riemannian hypersurfaces isometrically immersed in Riemannian or Lorentzian manifolds with constant sectional curvatures, highlighting the crucial role of CCVFs in the study of hypersurfaces. We apply these results to provide characterizations of compact Riemannian hypersurfaces in Euclidean space

R^{n + 1}

, Euclidean sphere

S^{n + 1}

, and de Sitter space

S_{1}^{n + 1}

. Full article

(This article belongs to the Special Issue Analysis on Differentiable Manifolds)

13 pages, 254 KiB

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

Viewed by 252

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

26 pages, 437 KiB

Open AccessFeature PaperArticle

Multivariate Polynomial Interpolation for Cubical Zero-Dimensional Schemes

by Edoardo Ballico

Axioms 2025, 14(4), 317; https://doi.org/10.3390/axioms14040317 - 21 Apr 2025

Viewed by 271

Abstract

We compute the multigraded Hilbert function of general unions of certain degree-8 zero-dimensional schemes, called 2cubes, for the Segre 3-folds and the three-dimensional smooth quadric hypersurface. On the Segre 3-folds, we handle the evaluation at the zero-dimensional scheme of all multigraded addenda of [...] Read more.

We compute the multigraded Hilbert function of general unions of certain degree-8 zero-dimensional schemes, called 2cubes, for the Segre 3-folds and the three-dimensional smooth quadric hypersurface. On the Segre 3-folds, we handle the evaluation at the zero-dimensional scheme of all multigraded addenda of their ring, not just the one associated to the Segre embedding. We discuss the Hilbert function and the index of regularity of the unions of several other low-degree zero-dimensional schemes. Full article

(This article belongs to the Section Algebra and Number Theory)

27 pages, 1140 KiB

Open AccessArticle

Singularity Analysis of Lightlike Hypersurfaces Generated by Null Cartan Curves in Minkowski Spacetime

by Xiaoming Fan, Yongsheng Zhu and Haijing Pan

Axioms 2025, 14(4), 279; https://doi.org/10.3390/axioms14040279 - 7 Apr 2025

Viewed by 365

Abstract

This study investigates the singularity structures of lightlike hypersurfaces generated by null Cartan curves in Minkowski spacetime. We construct a hierarchical geometric framework consisting of a lightlike hypersurface

L H_{β}

, a critical lightlike surface

L S_{β}

, and a degenerate [...] Read more.

This study investigates the singularity structures of lightlike hypersurfaces generated by null Cartan curves in Minkowski spacetime. We construct a hierarchical geometric framework consisting of a lightlike hypersurface

L H_{β}

, a critical lightlike surface

L S_{β}

, and a degenerate curve

L C_{β}

, with dimensions decreasing from 3D to 1D. Using singularity theory, we identify a novel geometric invariant

σ (t)

that governs the emergence of specific singularity types, including

C (2, 3) \times R^{2}

,

S W \times R

,

B F

,

C (B F)

,

C (2, 3, 4) \times R

, and

(2, 3, 4, 5)

-cusp. These singularities exhibit increasing degeneracy as the hierarchy progresses, with contact orders between the lightlike hyperplane

H S_{t_{0}}^{L}

and the curve

β

systematically intensifying. An explicit example demonstrates the construction of these objects and validates the theoretical results. This work establishes a systematic connection between null Cartan curves, stratified singularities, and contact geometry. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

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15 pages, 645 KiB

Open AccessArticle

Characterization of Finsler Space with Rander’s-Type Exponential-Form Metric

by Vinit Kumar Chaubey, Brijesh Kumar Tripathi, Sudhakar Kumar Chaubey and Meraj Ali Khan

Mathematics 2025, 13(7), 1063; https://doi.org/10.3390/math13071063 - 25 Mar 2025

Viewed by 369

Abstract

This study explores a unique Finsler space with a Rander’s-type exponential metric,

G (α, β) = (α + β) e^{\frac{β}{(α + β)}}

, where

α

is a Riemannian metric and

β

is a 1-form. [...] Read more.

This study explores a unique Finsler space with a Rander’s-type exponential metric,

G (α, β) = (α + β) e^{\frac{β}{(α + β)}}

, where

α

is a Riemannian metric and

β

is a 1-form. We analyze the conditions under which its hypersurfaces behave like hyperplanes of the first, second, and third kinds. Additionally, we examine the reducibility of the Cartan tensor

C

for these hypersurfaces, providing insights into their geometric structure. Full article

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

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15 pages, 335 KiB

Open AccessArticle

On the Secant Non-Defectivity of Integral Hypersurfaces of Projective Spaces of at Most Five Dimensions

by Edoardo Ballico

Symmetry 2025, 17(3), 454; https://doi.org/10.3390/sym17030454 - 18 Mar 2025

Viewed by 233

Abstract

Let

X \subset P^{n}

, where

3 \leq n \leq 5

, be an irreducible hypersurface of degree

d \geq 2

. Fix an integer

t \geq 3

. If

n = 5

, assume

t \geq 4

and [...] Read more.

Let

X \subset P^{n}

, where

3 \leq n \leq 5

, be an irreducible hypersurface of degree

d \geq 2

. Fix an integer

t \geq 3

. If

n = 5

, assume

t \geq 4

and

(t, d) \neq (4, 2)

. Using the Differential Horace Lemma, we prove that

O_{X} (t)

is not secant defective. For a fixed X, our proof uses induction on the integer t. The key points of our method are that for a fixed X, we only need the case of general linear hyperplane sections of X in lower-dimension projective spaces and that we do not use induction on d, allowing an interested reader to tackle a specific X for

n > 5

. We discuss (as open questions) possible extensions of some weaker forms of the theorem to the case where

n > 5

. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

21 pages, 3319 KiB

Open AccessArticle

Differential Geometry and Matrix-Based Generalizations of the Pythagorean Theorem in Space Forms

by Erhan Güler, Yusuf Yaylı and Magdalena Toda

Mathematics 2025, 13(5), 836; https://doi.org/10.3390/math13050836 - 2 Mar 2025

Viewed by 770

Abstract

In this work, we consider Pythagorean triples and quadruples using fundamental form matrices of hypersurfaces in three- and four-dimensional space forms and illustrate various figures. Moreover, we generalize that an immersed hypersphere

M^{n}

with radius r in an [...] Read more.

In this work, we consider Pythagorean triples and quadruples using fundamental form matrices of hypersurfaces in three- and four-dimensional space forms and illustrate various figures. Moreover, we generalize that an immersed hypersphere

M^{n}

with radius r in an

(n + 1)

-dimensional Riemannian space form

M^{n + 1} (c)

, where the constant sectional curvature is

c \in {- 1, 0, 1}

, satisfies the

(n + 1)

-tuple Pythagorean formula

P_{n + 1}

. Remarkably, as the dimension

n \to \infty

and the fundamental form

N \to \infty

, we reveal that the radius of the hypersphere converges to

r \to \frac{1}{\sqrt{2}}

. Finally, we propose that the determinant of the

P_{n + 1}

formula characterizes an umbilical round hypersphere satisfying

k_{1} = k_{2} = \dots = k_{n}

, i.e.,

H^{n} = K_{e}

in

M^{n + 1} (c)

. Full article

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

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14 pages, 280 KiB

Open AccessArticle

Spacelike Hypersurfaces in de Sitter Space

by Yanlin Li, Mona Bin-Asfour, Kholoud Saad Albalawi and Mohammed Guediri

Axioms 2025, 14(3), 155; https://doi.org/10.3390/axioms14030155 - 21 Feb 2025

Cited by 7 | Viewed by 750

Abstract

A closed conformal vector field in de Sitter space

S_{1}^{n + 1} (\bar{c})

induces a vector field on a spacelike hypersurface M of

S_{1}^{n + 1} (\bar{c})

, referred to as the induced vector field on M [...] Read more.

A closed conformal vector field in de Sitter space

S_{1}^{n + 1} (\bar{c})

induces a vector field on a spacelike hypersurface M of

S_{1}^{n + 1} (\bar{c})

, referred to as the induced vector field on M. This article investigates the characterization of compact spacelike hypersurfaces in de Sitter space without assuming the constancy of the mean curvature. Specifically, we establish that under certain conditions, a compact spacelike hypersurface in

S_{1}^{n + 1} (\bar{c})

is a sphere, that is, a totally umbilical hypersurface with constant mean curvature. We also present a different characterization of compact spacelike hypersurfaces in de Sitter space as spheres by using a lower bound on the integral of the Ricci curvature of the compact hypersurface in the direction of the induced vector field. We also consider de Sitter space as a Robertson–Walker space and provide several characterizations of spheres within its spacelike hypersurfaces. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Mathematical Physics)

30 pages, 2594 KiB

Open AccessArticle

Some New Geometric State-Space Properties of the Classical Linear Time-Optimal Control Problem with One Input and Real Non-Positive Eigenvalues of the System Following from Pontryagin’s Maximum Principle

by Borislav G. Penev

Axioms 2025, 14(2), 97; https://doi.org/10.3390/axioms14020097 - 28 Jan 2025

Viewed by 776

Abstract

This purely theoretical study considers two new geometric state-space properties of the classical linear time-optimal control problem with one input and real non-positive eigenvalues of the system, with constraints only on the control input and without constraints on the state-space variables, following from [...] Read more.

This purely theoretical study considers two new geometric state-space properties of the classical linear time-optimal control problem with one input and real non-positive eigenvalues of the system, with constraints only on the control input and without constraints on the state-space variables, following from Pontryagin’s maximum principle. These properties complement the well-known facts from the maximum principle about the number of switchings of the control function and the character of the optimal phase trajectories of the system leading it to the state-space origin. They lay the foundation of a new method for synthesizing the time-optimal control without the need to describe the switching hyper-surfaces. The new technique is demonstrated on two examples. The so-called “axes initialization” and the synthesis technique are illustrated on the double integrator system in its entirety. The second one is on a hypothetical seventh-order system. Full article

(This article belongs to the Special Issue Advances in Mathematical Methods in Optimal Control and Applications)

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28 pages, 399 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

On the Work of Cartan and Münzner on Isoparametric Hypersurfaces

by Thomas E. Cecil and Patrick J. Ryan

Axioms 2025, 14(1), 56; https://doi.org/10.3390/axioms14010056 - 13 Jan 2025

Cited by 1 | Viewed by 822

Abstract

A hypersurface

M^{n}

in a real space form

R^{n + 1}

,

S^{n + 1}

, or

H^{n + 1}

is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan [...] Read more.

A hypersurface

M^{n}

in a real space form

R^{n + 1}

,

S^{n + 1}

, or

H^{n + 1}

is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan and Münzner on the theory of isoparametric hypersurfaces in real space forms, in particular, spheres. This work is contained in four papers of Cartan published during the period 1938–1940 and two papers of Münzner that were published in preprint form in the early 1970s and as journal articles in 1980–1981. These papers of Cartan and Münzner have been the foundation of the extensive field of isoparametric hypersurfaces, and they have all been recently translated into English by T. Cecil. The paper concludes with a brief survey of the recently completed classification of isoparametric hypersurfaces in spheres. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

30 pages, 488 KiB

Open AccessFeature PaperArticle

Belyi Maps from Zeroes of Hypergeometric Polynomials

by Raimundas Vidunas

Mathematics 2025, 13(1), 156; https://doi.org/10.3390/math13010156 - 3 Jan 2025

Viewed by 765

Abstract

The evaluation of low-degree hypergeometric polynomials to zero defines algebraic hypersurfaces in the affine space of the free parameters and the argument of the hypergeometric function. This article investigates the algebraic surfaces defined by the hypergeometric equation [...] Read more.

The evaluation of low-degree hypergeometric polynomials to zero defines algebraic hypersurfaces in the affine space of the free parameters and the argument of the hypergeometric function. This article investigates the algebraic surfaces defined by the hypergeometric equation

{}_{2}F_{1} (- N, b; c; z) = 0

with

N = 3

or

N = 4

. As a captivating application, these surfaces parametrize certain families of genus 0 Belyi maps. Thereby, this article contributes to the systematic enumeration of Belyi maps. Full article

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21 pages, 395 KiB

Open AccessArticle

Interpolation of Polynomials and Singular Curves: Segre and Veronese Varieties

by Edoardo Ballico

Symmetry 2024, 16(12), 1683; https://doi.org/10.3390/sym16121683 - 19 Dec 2024

Viewed by 763

Abstract

We study an interpolation problem (objects singular at a prescribed finite set) for curves instead of hypersurfaces. We study singular curves in projective and multiprojective spaces. We construct curves that are singular (or with maximal dimension Zariski tangent space) at each point of [...] Read more.

We study an interpolation problem (objects singular at a prescribed finite set) for curves instead of hypersurfaces. We study singular curves in projective and multiprojective spaces. We construct curves that are singular (or with maximal dimension Zariski tangent space) at each point of a prescribed finite set, while the curves have low degree or low “complexity” (e.g., they are complete intersections of hypersurfaces of low degree). We discuss six open problems on the existence and structure of the base locus of the set of all hypersurfaces of a given degree and singular at a prescribed number of general points. The tools come from algebraic geometry, and some of the results are only existence ones or only asymptotic ones (but with as explicit as possible bounds). Some of the existence results are almost constructive, i.e., in our framework, random parameters should give a solution, or otherwise, take other random parameters. Full article

(This article belongs to the Section Mathematics)

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