Geometry

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 $MM$, that results from the integrability conditions of Killing’s equations. For $n=4$, the matrix $MM$ is of size $20\times 6$. In the case where $MM$ 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 $MM$ 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.

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.

It was known in antiquity that the sum of the three angles of a triangle equals $\pi$. Surprisingly, it was not until 1952 that the corresponding question for a tetrahedron was addressed. In that year, J.W. Gaddum proved that the sum of the four solid angles in a tetrahedron lies within the interval of and those lower and upper bounds are the best possible. In 2020, H. Katsuura showed that $2\pi$ was unachievable. In this paper, we generalize these results to show that for a non-degenerate n-simplex in ${\mathbb{R}}^n$ with $n\ge 3$, the solid angles at the vertices add up to a positive number that is less than one-half the -dimensional area of the unit sphere in ${\mathbb{R}}^n$. We also show that there are examples for which the sum can be made arbitrarily close to the extreme values of 0 and one-half the -dimensional area of the unit sphere in ${\mathbb{R}}^n$.