Geometry

22 pages, 1110 KB

Open AccessArticle

by Yevgenya Movshovich and John Wetzel

Geometry 2025, 2(4), 18; https://doi.org/10.3390/geometry2040018 - 4 Nov 2025

In recent years, two quite different tools have been employed to study global properties of arcs in the plane. The first is drapeability, which grew from ideas of J. R. Alexander in early 2000s defining an arc drapeable if it lies in the [...] Read more.

Λ

-configuration, where an arc travels from one line to another and back. We investigate interrelations between these notions and in the process find drapeability criteria for open arcs, necessary and sufficient drapeability conditions for three-segment z-shaped arcs, and new bounds for the width of non-drapeable arcs. Full article

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

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32 pages, 355 KB

Open AccessArticle

Killing Vector Fields of Invariant Metrics

by Gerard Thompson

Geometry 2025, 2(4), 17; https://doi.org/10.3390/geometry2040017 - 22 Oct 2025

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Abstract

We study the existence of Killing vector fields for right-invariant metrics on low-dimensional Lie groups. Specifically, Lie groups of dimension two, three and four are considered. Before attempting to implement the differential conditions that comprise Killing’s equations, the metric is reduced as much as possible by using the automorphism group of the Lie algebra. After revisiting the classification of the low-dimensional Lie algebras, we review some of the known results about Killing vector fields on Lie groups and add some new observations. Then we investigate indecomposable Lie algebras and attempt to solve Killing’s equations for each reduced metric. We introduce a matrix

M M

, that results from the integrability conditions of Killing’s equations. For

n = 4

, the matrix

M M

is of size

20 \times 6

. In the case where

M M

has maximal rank, for the Lie group problem considered in this article, only the left-invariant vector fields are Killing. The solution of Killing’s equations is performed by using MAPLE, and knowledge of the rank of

M M

can help to confirm that the solutions found by MAPLE are the only linearly independent solutions. After finding a maximal set of linearly independent solutions, the Lie algebra that they generate is identified to one in a standard list. Full article

38 pages, 5294 KB

Open AccessArticle

A Discrete Schwarzian Derivative via Circle Packing

by Kenneth Stephenson

Geometry 2025, 2(4), 16; https://doi.org/10.3390/geometry2040016 - 9 Oct 2025

Viewed by 341

Abstract

There exists an extensive and fairly comprehensive discrete analytic function theory which is based on circle packing. This paper introduces a faithful discrete analogue of the classical Schwarzian derivative to this theory and develops its basic properties. The motivation comes from the current lack of circle packing algorithms in spherical geometry, and the discrete Schwarzian derivative may provide for new approaches. A companion localized notion called an intrinsic schwarzian is also investigated. The main concrete results of the paper are limited to circle packing flowers. A parameterization by intrinsic schwarzians is established, providing an essential packing criterion for flowers. The paper closes with the study of special classes of flowers that occur in the circle packing literature. As usual in circle packing, there are pleasant surprises at nearly every turn, so those not interested in circle packing theory may still enjoy the new and elementary geometry seen in these flowers. Full article

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Geometry, Volume 2, Issue 4 (December 2025) – 3 articles

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