Journal Description

Geometry

Geometry is an international, peer-reviewed, open access journal which provides an advanced forum for studies related to geometry, and is published quarterly online by MDPI.

Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
Rapid Publication: first decisions in 18 days; acceptance to publication in 7 days (median values for MDPI journals in the first half of 2026).
Recognition of Reviewers: Reviewers whose reports are timely and of high quality receive an APC discount voucher for a future publication in an MDPI journal. Become a reviewer.
Geometry is a companion journal of Mathematics.
Journal Cluster of Mathematics and Its Applications: AppliedMath, Axioms, Computation, Fractal and Fractional, Geometry, International Journal of Topology, Logics, Mathematics and Symmetry.

Imprint Information Journal Flyer Open Access ISSN: 3042-402X

Latest Articles

54 pages, 503 KB

Open AccessArticle

Twisted Kodaira–Spencer Classes and the Inequality of Severi

by Igor Reider

Geometry 2026, 3(3), 17; https://doi.org/10.3390/geometry3030017 - 3 Sep 2026

Abstract

Let X be a smooth minimal complex surface of general type subject to the inequality

K_{X}^{2} < 4 χ (O_{X}) .

The old problem of Severi states that the image of the Albanese map of such a surface [...] Read more.

Let X be a smooth minimal complex surface of general type subject to the inequality

K_{X}^{2} < 4 χ (O_{X}) .

The old problem of Severi states that the image of the Albanese map of such a surface is at most a curve. The problem was completely settled by Rita Pardini using the works of Xiao on fibered surfaces. The paper suggests a different approach to Severi type inequalities through the study of the cohomology group

H^{1} (Θ_{X} (- K_{X}))

. For X irregular, with no irrational pencil, it is shown that the cohomology classes of the group carry geometric data:

P (H^{1} (Θ_{X} (- K_{X}))) ∋ [ξ] \mapsto (F_{ξ}, σ_{ξ}, A_{ξ})

where

F_{ξ}

is a rank 2 bundle,

σ_{ξ} : F_{ξ} ⟶ Ω_{X}

is a modification of

Ω_{X}

along an effective, nonzero divisor

E_{ξ}

inducing the isomorphism on the level of global sections:

H^{0} (σ_{ξ}) : H^{0} (F_{ξ}) ≅ H^{0} (Ω_{X});

A_{ξ}

is at most 0-dimensional subscheme of

E_{ξ}

. The paper establishes many geometrical properties of

E_{ξ}

and

A_{ξ}

. As a consequence, for an irregular surface X without irrational pencil, sufficient conditions are given for a Severi type inequality

2 K_{X}^{2} \geq c_{2} + q - 2,

where

(K_{X}^{2}, c_{2})

and

q = h^{0} (Ω_{X})

are the Chern numbers and the irregularity of X, respectively. The dependence of the geometric data

(F_{ξ}, σ_{ξ}, A_{ξ})

[ξ]

is explored: it is shown that there is a finite collection of projective subspaces

{P (W_{E})}

P (H^{1} (Θ_{X} (- K_{X})))

, labeled by effective nonzero divisors E; the data are constant on the complement of the collection and undergoes a change on each

P (W_{E})

. One attaches a quiver to this collection which is an interesting object to study. Full article

45 pages, 602 KB

Open AccessReview

Remarks on Symplectic Geometry

by Jae-Hyun Yang

Geometry 2026, 3(3), 16; https://doi.org/10.3390/geometry3030016 - 24 Aug 2026

Abstract

The relatively new subject, symplectic geometry, has been studied the past five decades. Symplectic geometry is the mathematical subject studying the geometry of symplectic manifolds. A symplectic manifold is an even-dimensional smooth manifold equipped with a closed non-degenerate two form. This paper surveys the progress on the study of symplectic geometry during the past five decades. It deals with a brief history of symplectic geometry and symplectic topology, Arnold’s conjecture, pseudoholomorphic curves, the convexity properties of a moment map, recent results on Hamiltonian and non-Hamiltonian symplectic group actions, the classification of symplectic group actions, recent results on the symplectic embedding problems, the theory of Gromov-Witten invariants, Lagrangian Floer homology, the Fukaya category, and mirror symmetry. Full article

13 pages, 893 KB

Open AccessArticle

Apollonius and the Hyperbolic Circle

by Andrew J. Simoson

Geometry 2026, 3(3), 15; https://doi.org/10.3390/geometry3030015 - 7 Aug 2026

Abstract

Given a positive number

n > 1

and two distinct planar focal points A and B, the locus of all points P such that

| A P | = n | B P |

is a circle of Apollonius of index n [...] Read more.

Given a positive number

n > 1

and two distinct planar focal points A and B, the locus of all points P such that

| A P | = n | B P |

is a circle of Apollonius of index n Under what conditions is B the center of a hyperbolic circle in the Poincaré disk? In particular, for any circle

E

C

lying within the unit circle, where the Euclidean center u of

E

is other than O and its Euclidean radius is

ϵ

0 < ϵ < 1

, there exists another circle

D

and a point B in

E

where the center of

D

is O and its Euclidean radius is

δ

0 < δ < 1

, for which

D

and

E

are hyperbolic translates of one another, as are O and B; by rotation symmetry of

C

about O, we may take A, B, and u as real numbers with

v = B

, and

u + ϵ n = A

, with

0 < u < v < 1

, and we conclude that

n δ v = 1

. That is, B is the hyperbolic center of

E

as well as the inner Apollonian focus of

E

whose Apollonian index is n, and

A = u + ϵ n

is the outer focus of

E

. Furthermore, if P and Q are distinct points on such a circle of Apollonius with index n we have the cross-ratio of the four points (not necessarily collinear) for which

\frac{| A P | | B Q |}{| A Q | | B P |} = 1

; with X being an indeterminate point, this identity in turn means that

\frac{| A P | | B X |}{| A Q | | B X |} = \frac{| A P | | B Q |}{| A Q | | B P |}

is the equation of a circle, demonstrating why this old Apollonian algorithm forms the core idea for a hyperbolic metric. Full article

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32 pages, 455 KB

Open AccessArticle

The Fibonacci Space-Filling Curve

by Mustafa İsmail Özkaraca

Geometry 2026, 3(3), 14; https://doi.org/10.3390/geometry3030014 - 20 Jul 2026

Abstract

Can you stretch and reform a curve such that it fills a square completely? This question dates back to the late 19th century—the origin of space-filling curves. It was proven affirmatively by many great mathematicians. In this document, we reconsider the problem and present a different proof using the Cartesian product of Fibonacci substitution with itself. Our construction differs from other curves by the particular ordering produced by the Cartesian product of Fibonacci substitutions and the iterative geometric design induced by it. Full article

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36 pages, 474 KB

Open AccessArticle

Nearapproaching, a First Presentation

by Dieter Leseberg and Zohreh Vaziry

Geometry 2026, 3(3), 13; https://doi.org/10.3390/geometry3030013 - 1 Jul 2026

Abstract

We consider nearapproaching, a simple generalisation of pseudonearness and approach distances. The central idea in an approach space

(X, d)

in the sense of Lowen is that of a distance d, which is a function on [...] Read more.

We consider nearapproaching, a simple generalisation of pseudonearness and approach distances. The central idea in an approach space

(X, d)

in the sense of Lowen is that of a distance d, which is a function on

X \times 2^{X}

[0, \infty]

. Of fundamental importance is the fact that such a distance can be defined not only in metric spaces, but also in topological spaces, uniform spaces, and related structures. Pseudonearness structures on a set X consist of pairs

(B^{X}, N)

, where

B^{X}

is a bornology and N is a nearoperator from

B^{X}

\underset{̲}{P} (\underset{̲}{P} (\underset{̲}{P} X))

, satisfying certain near axioms. In this framework, bornologies, generalised Kuratowski operators, pseudoproximity structures, and, last but not least, classical nearness spaces can all be unified within a single setting. In our treatment of approaching, we define the corresponding spaces by focusing on functions

τ

from

C X \subset \underset{̲}{P} (\underset{̲}{P} (X)

into the set of functions from

B^{X}

into the closed interval

[0, \infty]

, known as targets, which we compare setwise, where

B^{X}

is a boundedness structure. Thus, a target can be regarded as a map that measures how close a collection

S \in C X

is to a bounded set

B \in B^{X}

. Full article

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14 pages, 913 KB

Open AccessFeature PaperArticle

Information Geometry Description of Inferential Scattering

by Marco Favretti

Geometry 2026, 3(3), 12; https://doi.org/10.3390/geometry3030012 - 30 Jun 2026

Abstract

We investigate the geometrical structure underlying the notion of inferential scattering, which was formulated by E. T. Jaynes in the 1980s using the language of equilibrium statistical mechanics. We show that inferential scattering can be naturally defined on a dually flat Riemannian manifold equipped with dual coordinate systems, a differential-geometric structure that occupies a central place in Information Geometry. We find that the controlled evolution of the system on the dually flat manifold can be expressed as the horizontal lift of an integrable connection. Full article

(This article belongs to the Topic Geometric Structures on Manifolds: Riemannian Geometry, Submanifolds, and Physical Applications)

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25 pages, 838 KB

Open AccessFeature PaperArticle

Central Conics in

H^{2}

Are Fibers over the Group of Steiner Conics

by John Sarli

Geometry 2026, 3(2), 11; https://doi.org/10.3390/geometry3020011 - 11 Jun 2026

Abstract

We provide an intrinsic construction of the central conics in the real hyperbolic plane

H^{2}

, whereby each conic C is the composition of a unique pair of Steiner conics (those generated by collineations). The composition is achieved by elliptic curve addition [...] Read more.

We provide an intrinsic construction of the central conics in the real hyperbolic plane

H^{2}

, whereby each conic C is the composition of a unique pair of Steiner conics (those generated by collineations). The composition is achieved by elliptic curve addition on intersection points of the two components with their orthogonal trajectories, which have a natural representation as genus 1 curves in any inversive model of

H^{2}

. The central Steiner conics that have a focal axis L are identified with the subgroup

G (L)

of collineations generated by reflections in the lines perpendicular to L. We obtain a

G (L)

-equivariant partition of the central conics by defining the fiber over

g \in G (L)

to be the set of compositions C such that

π (C) = g

. Here,

π (C)

is the unique Steiner conic tangent to C at the points on L, and is the product of the two elements in

G (L)

that represent the components of C. We use the terminology of fibers strictly in an incidence-geometric sense. Full article

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10 pages, 813 KB

Open AccessFeature PaperArticle

Orthogonal 2-Sphere Basis of Stable 4-Sphere

by Akio Kawauchi

Geometry 2026, 3(2), 10; https://doi.org/10.3390/geometry3020010 - 19 May 2026

Abstract

Every stable 4-sphere is identified with the double branched covering space of a trivial surface-knot space. As a result of Wall, it is known that any two orthogonal bases of every stable 4-sphere are transformed into each other by an orientation-preserving diffeomorphism of the stable 4-sphere. In this paper another proof of Wall’s result is presented, and strengthened in the sense that the lift of an equivalence of the trivial surface-knot space can be taken as the diffeomorphism. Two applications are made. The first shows that every orientation-preserving diffeomorphism of every stable 4-sphere is nothing but the double branched covering lift of an equivalence of a trivial surface-knot space up to a smooth isotopy and a composition with an identity-shift. The second gives a similar result for TOP stable 4-spheres. Here, even if it is a smooth 4-manifold, unless it is diffeomorphic to the stable 4-sphere, the TOP trivial surface-knot space cannot be smooth. Full article

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29 pages, 2822 KB

Open AccessFeature PaperArticle

Wessel’s Algebra and Morley’s Theorem

by Sebastian Xambó-Descamps

Geometry 2026, 3(2), 9; https://doi.org/10.3390/geometry3020009 - 8 May 2026

Abstract

This paper is devoted to provide a proof of F. Morley’s theorem concerning triangles in the Euclidean plane

E

(see Theorem 1 in the Introduction section) phrased in terms of the geometric algebra

G

E

(called Wessel’s algebra). This algebra is studied [...] Read more.

This paper is devoted to provide a proof of F. Morley’s theorem concerning triangles in the Euclidean plane

E

(see Theorem 1 in the Introduction section) phrased in terms of the geometric algebra

G

E

(called Wessel’s algebra). This algebra is studied in detail in Section 2, its uses in describing isometries of

E

in Section 3, its bearing on the geometry of Morley’s construction in Section 4, and the claimed proof in Section 5. Morley’s theorem can be extended by using all the trisectors (interior and exterior) of a triangle, and suitable intersections of them. These intersections form what we call Morley’s constellation and out of it 36 generalized Morley triangles can be formed. Among these triangles, 27 are equilateral and with sides parallel to the original Morley triangle (Appendix B). The 36 triangles are depicted in Appendix C. All graphics in this work have been created by the author. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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14 pages, 300 KB

Open AccessFeature PaperArticle

Translative Covering a Square with Isosceles Right Triangles

by Janusz Januszewski and Łukasz Zielonka

Geometry 2026, 3(2), 8; https://doi.org/10.3390/geometry3020008 - 13 Apr 2026

Abstract

A translative covering the rectangle

a \times b

with homothetic copies of a right isosceles triangle T (of the legs parallel to the sides of

a \times b

) is considered. It is shown that any collection of equal triangles homothetic to T [...] Read more.

A translative covering the rectangle

a \times b

with homothetic copies of a right isosceles triangle T (of the legs parallel to the sides of

a \times b

) is considered. It is shown that any collection of equal triangles homothetic to T with the total area at least 2 permits a translative covering of

1 \times 1

; this bound is tight. It is also demonstrated that any collection of positive homothetic copies of T with the total area at least 3 permits a translative covering of

1 \times 1

. Moreover, it is proven that if

a \geq \frac{5 + \sqrt{33}}{4} b

, then any collection of triangles homothetic to T with the total area at least

\frac{1}{2} {(a + b)}^{2}

permits a translative covering of

a \times b

; this bound is tight. Full article

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15 pages, 285 KB

Open AccessArticle

Pseudoconvexity and Steinness of Connected Complex Lie Groups: A Concise Lie-Theoretic Approach

by Abdel Rahman Al-Abdallah

Geometry 2026, 3(2), 7; https://doi.org/10.3390/geometry3020007 - 1 Apr 2026

Abstract

We give new concise Lie-theoretic proofs of basic analytic–geometric properties of connected complex Lie groups. Using Matsushima’s biholomorphic splitting

G ≃ C^{n} \times \tilde{K}

together with a refined analysis of the center via its Cousin factor, we show that every connected [...] Read more.

We give new concise Lie-theoretic proofs of basic analytic–geometric properties of connected complex Lie groups. Using Matsushima’s biholomorphic splitting

G ≃ C^{n} \times \tilde{K}

together with a refined analysis of the center via its Cousin factor, we show that every connected complex Lie group is pseudoconvex. Our approach is structural: it reduces to the reductive factor, separates the semisimple and central parts, and concludes using permanence of pseudoconvexity under products and finite quotients, together with standard triviality results for holomorphic principal bundles over Stein bases. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

12 pages, 281 KB

Open AccessFeature PaperArticle

Geometric Reinterpretation of Partial Differential Equations and Applications

by Satyanad Kichenassamy

Geometry 2026, 3(1), 6; https://doi.org/10.3390/geometry3010006 - 13 Mar 2026

Abstract

We obtain improved regularity estimates on solutions of partial differential equations by combining the method of Fuchsian Reduction with geometric transformations. Examples include the meron problem and the regularity of the conformal radius. In each case, Reduction needs to be combined with a reinterpretation of the underlying geometry. We argue that the geometric meaning assigned to a problem has an influence, positive or negative, on the range of methods envisioned for its solution, and that the Euler–Poisson–Darboux (EPD) equation cannot be properly understood within a single geometric framework. This explains the central position of EPD-like equations. Full article

10 pages, 299 KB

Open AccessArticle

Focal Circular Cubic and Complex Numbers

by Jiří Blažek

Geometry 2026, 3(1), 5; https://doi.org/10.3390/geometry3010005 - 10 Mar 2026

Abstract

A focal circular cubic is a locus of the foci of conics tangent to a given quadrilateral. In this article, we derive some of the focal curve’s basic properties. We study the curve in the complex plane and prove that the complex coordinates of pairs of foci satisfy a quadratic equation. This equation can be expressed as a linear combination of two basic quadratic equations, which form a basis of a vector space. Furthermore, we give a nonstandard analytical condition, expressed in complex numbers, under which a circle can be inscribed in a quadrilateral. Finally, we leave the complex plane and show the construction of an arbitrary pair of foci of the curve by Euclidean means. Full article

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14 pages, 438 KB

Open AccessArticle

Properties of Elliptic Cycloids

by Matthew A. Pons and Nicholas D. White

Geometry 2026, 3(1), 4; https://doi.org/10.3390/geometry3010004 - 9 Feb 2026

Abstract

Given an ellipse rolling along a straight line without slipping and a point P on the ellipse, we will determine the shape of the elliptic cycloid traced by P as the ellipse rolls and compute the area under one arch of the elliptic [...] Read more.

(This article belongs to the Special Issue Feature Papers in Geometry)

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13 pages, 577 KB

Open AccessFeature PaperArticle

Family of Non-Minkowski Measurable Fractals in

R^{2}

by Uta Freiberg and Jonas Lippold

Geometry 2026, 3(1), 3; https://doi.org/10.3390/geometry3010003 - 2 Feb 2026

Abstract

A long-standing conjecture of Lapidus asserts that under certain conditions a self-similar fractal set is not Minkowski measurable if and only if it is of lattice-type. For self-similar sets in

R

, the Lapidus conjecture has been confirmed. However, in higher dimensions, it [...] Read more.

A long-standing conjecture of Lapidus asserts that under certain conditions a self-similar fractal set is not Minkowski measurable if and only if it is of lattice-type. For self-similar sets in

R

, the Lapidus conjecture has been confirmed. However, in higher dimensions, it remains unclear whether all lattice-type self-similar sets are not Minkowski measurable. This work presents a family of lattice-type subsets in

R^{2}

that are not Minkowski measurable, hence providing further support for the conjecture. Furthermore, an argument is presented to illustrate why these sets are not covered by previous results. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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15 pages, 302 KB

Open AccessArticle

Witten Deformation and Divergence-Free Symmetric Killing 2-Tensors

by Kwangho Choi and Junho Lee

Geometry 2026, 3(1), 2; https://doi.org/10.3390/geometry3010002 - 13 Jan 2026

Cited by 1

Abstract

By using a Morse function and a Witten deformation argument, we obtain an upper bound for the dimensions of the space of divergence-free symmetric Killing p-tensors on a closed Riemannian manifold and explicitly calculate it for

p = 2

. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

13 pages, 1033 KB

Open AccessArticle

A New Characterization of the Symmedian Point

by Yu Chen and Robert J. Fisher

Geometry 2026, 3(1), 1; https://doi.org/10.3390/geometry3010001 - 8 Jan 2026

Abstract

Let

▵ A B C

be a triangle in the plane

E

, K be its symmedian point, and

C

be its circumcircle. Assume that P is a point on

C

such that it is not A, B, or C, [...] Read more.

Let

▵ A B C

be a triangle in the plane

E

, K be its symmedian point, and

C

be its circumcircle. Assume that P is a point on

C

such that it is not A, B, or C, it does not lie on the medians of

▵ A B C

, and the lines

ℓ_{A P}

ℓ_{B P}

, and

ℓ_{C P}

intersect

ℓ_{B C}

ℓ_{C A}

, and

ℓ_{A B}

at points

P_{a}

P_{b}

, and

P_{c}

, respectively. By the Ceva Concurrence Theorem, the harmonic conjugates

Q_{a}

Q_{b}

, and

Q_{c}

P_{a}

P_{b}

, and

P_{c}

\bar{B C}

\bar{C A}

, and

\bar{A B}

, respectively, are collinear. We prove that K lies on the line through

Q_{a}

Q_{b}

, and

Q_{c}

, which provides a new characterization of the symmedian point. Moreover, this one-to-one correspondence extends to a bijection from the entire circumcircle

C

onto the set

L_{K}

of all the lines in

E

that pass through K. Full article

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15 pages, 424 KB

Open AccessArticle

Some Nice Configurations of Golden Triangles

by Aldo Scimone

Geometry 2025, 2(4), 21; https://doi.org/10.3390/geometry2040021 - 10 Dec 2025

Abstract

It is well known among geometry scholars that the golden triangle, an isosceles triangle with sides and base in golden ratio, maintains a significant relationship with regular polygons, notably the regular pentagon, pentagram, and decagon. Extensive mathematical literature addresses this subject. Furthermore, its close association with the golden ratio—a mathematical concept describing a harmonious and proportionate relationship between segments—renders it a noteworthy element in the fields of geometry, art, and architecture. Nevertheless, the interrelationships among these mathematical constructs frequently reveal unexpected configurations, thereby accentuating intriguing patterns. The purpose of this investigation is to highlight these novel configurations, which indicate new connections between the golden triangle and regular polygons. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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8 pages, 217 KB

Open AccessArticle

A Simple Proof of Chen–Ricci Inequality and Applications

by Ion Mihai

Geometry 2025, 2(4), 20; https://doi.org/10.3390/geometry2040020 - 1 Dec 2025

Abstract

In the present paper, we give a simple proof of the Chen–Ricci inequality for submanifolds in Riemannian and Lorentzian space forms, respectively. Moreover, we extend the Chen–Ricci inequality to submanifolds in Lorentzian manifolds with a semi-symmetric non-metric connection. Full article

14 pages, 318 KB

Open AccessArticle

Side-Angle-Side, Side-Angle-Angle, and the Non-Continuous Absolute Plane

by John Donnelly

Geometry 2025, 2(4), 19; https://doi.org/10.3390/geometry2040019 - 17 Nov 2025

Abstract

We prove that if we start with a non-continuous absolute plane, remove Side-Angle-Side as an axiom, and replace it with the three new axioms, Side-Angle-Angle, angle addition, and the existence of angle bisectors, then the result is also an absolute plane. Full article

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