Geometry

43 pages, 497 KiB

Open AccessArticle

Problems in Invariant Differential Operators on Homogeneous Manifolds

by Jae-Hyun Yang

Geometry 2025, 2(2), 9; https://doi.org/10.3390/geometry2020009 - 9 Jun 2025

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In this paper, we consider six homogeneous manifolds

G L (n, R) / O (n, R)

S L (n, R) / S O (n, R)

, [...] Read more.

In this paper, we consider six homogeneous manifolds

G L (n, R) / O (n, R)

S L (n, R) / S O (n, R)

S p (2 n, R) / U (n),

(G L (n, R) ⋉ R^{(m, n)}) / O (n, R)

(S L (n, R) ⋉ R^{(m, n)}) / S O (n, R),

(S p (2 n, R) ⋉ H_{R}^{(n, m)}) / (U (n) \times S (m, R)) .

They are homogeneous manifolds which are important geometrically and number theoretically. These first three spaces are well-known symmetric spaces and the other three are not symmetric spaces. It is well known that the algebra of invariant differential operators on a symmetric space is commutative. The algebras of invariant differential operators on these three non-symmetric spaces are not commutative and have complicated generators. We discuss invariant differential operators on these non-symmetric spaces and provide natural but difficult problems about invariant theory. Full article

10 pages, 304 KiB

Open AccessArticle

On the Relation Between a Locus and Poncelet’s Closure Theorem

by Jiří Blažek

Geometry 2025, 2(2), 8; https://doi.org/10.3390/geometry2020008 - 9 Jun 2025

Viewed by 200

Abstract

This article contains a synthetic proof of the following proposition: consider a conic

c_{1}

and its variable chord

A B

, which subtends a right angle at a given point P. Then, the foot E of the perpendicular dropped from P [...] Read more.

This article contains a synthetic proof of the following proposition: consider a conic

c_{1}

and its variable chord

A B

, which subtends a right angle at a given point P. Then, the foot E of the perpendicular dropped from P onto the line

A B

lies on a certain circle (the line being the limiting case of the circle). To prove this proposition, we show how Poncelet’s closure theorem for quadrilaterals can be derived by elementary projective considerations only (without any computations, either in Cartesian or projective coordinates). Finally, the limiting case of the proposition, where the point P lies on the conic, is also mentioned. The problem can then be reduced to Frégier’s theorem and may represent a slightly different perspective on this theorem. Full article

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

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27 pages, 1329 KiB

Open AccessFeature PaperArticle

Defining and Visualizing the Geometry of Relativistic Physics

by Yaakov Friedman and Tzvi Scarr

Geometry 2025, 2(2), 7; https://doi.org/10.3390/geometry2020007 - 14 May 2025

Viewed by 285

Abstract

We continue Riemann’s program of geometrizing physics, extending it to encompass gravitational and electromagnetic fields as well as media, all of which influence the geometry of spacetime. The motion of point-like objects—both massive and massless—follows geodesics in this modified geometry. To describe this geometry, we generalize the notion of a metric to local scaling functions which permit not only quadratic but also linear dependence on temporal and spatial separations. Our local scaling functions are defined on flat spacetime coordinates. We demonstrate how to construct various geometries directly from field sources, using symmetry and superposition, without relying on field equations. For each geometry, two key visualizations elucidate the connection between the geometry and the dynamics as follows: (1) the cross-sections of the ball of admissible velocities, and (2) the cross-sections of the local scaling function. Full article

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

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12 pages, 7101 KiB

Open AccessFeature PaperArticle

Hyperbolic Cords and Wheels

by Andrew J. Simoson

Geometry 2025, 2(2), 6; https://doi.org/10.3390/geometry2020006 - 6 May 2025

Viewed by 238

Abstract

The cycloidal family of curves in

R^{2}

The cycloidal family of curves in

R^{2}

, also known as the trochoid family, are equivalently generated by two classic methods: bungee cords and rolling wheels. What about their counterpart families in the hyperbolic unit disk? We review the two methods in Euclidean space, outline pertinent hyperbolic geometry tools, using both the Klein and Poincaré models, and show that the two methods give distinct, yet similar, results in hyperbolic space. Full article

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

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15 pages, 802 KiB

Open AccessArticle

A Theoretical Framework for Computing Generalized Weighted Voronoi Diagrams Based on Lower Envelopes

by Martin Held and Stefan de Lorenzo

Geometry 2025, 2(2), 5; https://doi.org/10.3390/geometry2020005 - 17 Apr 2025

Viewed by 329

Abstract

This paper presents a theoretical framework for constructing generalized weighted Voronoi diagrams (GWVDs) of weighted points and straight-line segments (“sites”) in the Euclidean plane, based on lower envelopes constructed in three-dimensional space. Central to our approach is an algebraic distance function that defines the minimum weighted distance from a point to a site. We also introduce a parameterization for the bisectors, ensuring a precise representation of Voronoi edges. The connection to lower envelopes allows us to derive (almost tight) bounds on the combinatorial complexity of a GWVD. We conclude with a short discussion of implementation strategies, ranging from leveraging computational geometry libraries to employing graphics hardware for approximate solutions. Full article

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25 pages, 368 KiB

Open AccessArticle

LU Factorizations for ℕ × ℕ-Matrices and Solutions of the k[S]-Hierarchy and Its Strict Version

by G. F. Helminck and J. A. Weenink

Geometry 2025, 2(2), 4; https://doi.org/10.3390/geometry2020004 - 15 Apr 2025

Viewed by 221

Abstract

Let S be the

N \times N

-matrix of the shift operator and let k denote the field of real or complex numbers. We consider two different deformations of the commutative algebra

k [S]

, together with the evolution equations of [...] Read more.

Let S be the

N \times N

-matrix of the shift operator and let k denote the field of real or complex numbers. We consider two different deformations of the commutative algebra

k [S]

, together with the evolution equations of the deformations of the powers

{S^{i}, i ⩾ 1}

. They are called the

k [S]

-hierarchy and the strict

k [S]

-hierarchy. For suitable Banach spaces B, we explain how LU factorizations in

GL (B)

can be used to produce dressing matrices of both hierarchies. These dressing matrices correspond to bounded operators on B, a class far more general than the one used at a prior construction. This wider class of solutions of both hierarchies makes it possible to treat reductions of both systems. The matrix coefficients of these matrices are shown to be quotients of analytic functions. Moreover, we present a subgroup

G_{c p t} (B)

GL (B)

such that the procedure with LU factorizations works for each

g \in G_{c p t} (B)

. Full article

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

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Geometry, Volume 2, Issue 2 (June 2025) – 6 articles

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