Geometry and Topology Commons^™

How To Find Killing Vectors, Charles G. Torre

How to... in 10 minutes or less

We show how to compute the Lie algebra of Killing vector fields of a metric in Maple using the commands KillingVectors and LieAlgebraData. A Maple worksheet and a PDF version can be found below.

The Octonions And The Exceptional Lie Algebra G2, Ian M. Anderson

Research Vignettes

The octonions O are an 8-dimensional non-commutative, non-associative normed real algebra. The set of all derivations of O form a real Lie algebra. It is remarkable fact, first proved by E. Cartan in 1908, that the the derivation algebra of O is the compact form of the exceptional Lie algebra G2. In this worksheet we shall verify this result of Cartan and also show that the derivation algebra of the split octonions is the split real form of G2.

PDF and Maple worksheets can be downloaded from the links below.

A Homogeneous Solution Of The Einstein-Maxwell Equations, Charles G. Torre

Research Vignettes

We exhibit and analyze a homogeneous spacetime whose source is a pure radiation electromagnetic field [1]. It was previously believed that this spacetime is the sole example of a homogeneous pure radiation solution of the Einstein equations which admits no electromagnetic field (see [2] and references therein). Here we correct this error in the literature by explicitly displaying the electromagnetic source. This result implies that all homogeneous pure radiation spacetimes satisfy the Einstein-Maxwell equations.

PDF and Maple worksheets can be downloaded from the links below.

How To Create A Lie Algebra, Ian M. Anderson

How to... in 10 minutes or less

We show how to create a Lie algebra in Maple using three of the most common approaches: matrices, vector fields and structure equations. PDF and Maple worksheets can be downloaded from the links below.

Infinite Product Spaces Under The Tychonoff And Goofynoff Topologies, James A. Capps

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The only topology considered for the infinite product of topological spaces in most current topology texts and research papers is the Tychonoff topology. Yet there is another topology which seems to be a much more topologically natural generalization of the usual "box" topology of finite products. We call this natural generalization the Goofynoff topology and exploit its properties. The use of the word "Goofynoff" (pronounced Goof'-n-off) is not universal and does not refer to any person of that name. In the few references to this topology that can be found, it is usually called simply the Box Topology. None of …

The Fundamental Groups Of The Complements Of Some Solid Horned Spheres, Norman William Riebe

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

One of the methods used for the construction of the classical Alexander horned sphere leads naturally to generalization to horned spheres of higher order. Let M₂, denote the Alexander horned sphere. This is a 2-horned sphere of order 2. Denote by M₃ and M₄, two 2-horned spheres of orders 3 and 4, respectively, constructed by such a generalization.

The fundamental groups of the complements of M₂, M₃, and M₄ are derived, and representations of these groups onto the Alternating Group, A₅, are found. The form of the presentations …

An Investigation Of The Properties Of Join Geometry, Louis John Giegerich Jr.

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This paper presents a proof that the classical geometry as stated by Karol Borsuk [1] follows from the join geometry of Walter Prenowitz [2].

The approach taken is to assume the axioms of Prenowitz. Using these as the foundation, the theory of join geometry is then developed to include such ideas as 'convex set', 'linear set', the important concept of 'dimension', and finally the relation of 'betweenness'. The development is in the form of definitions with the important extensions given in the form of theorems.

With a firm foundation of theorems in the join geometry, the axioms of classical geometry …