Ms Chaitanya Shettigara

Honours thesis

Symmetries on Manifolds

Symmetry is intuitively a simple concept. It is one of the first topics taught in mathematics; the first shapes seen, such as circles, triangles and squares are all highly symmetric. It is associated with beauty and balance. It also has practical applications in problem solving. In physics, symmetry is very important as it is associated with invariance and conservation laws. Sometimes symmetries are clear; for example, it does not take high powered calculation to recognise that the surface of a sphere is very symmetric. Othertimes, it is not so clear, such as in more abstract spaces. This begs the question, how do we mathematically describe and find symmetries? Using the 2-sphere as a motivating example, this thesis seeks to answer this question. An analysis of this topic requires preliminary knowledge, for example, how do we describe a space? The answer is through manifolds, which is the realm of differential geometry. In turn, this is based on an understanding of topological spaces. The first chapters of this thesis lay a foundation of these concepts, including defining a manifold, and also developing the Riemannian metric, which is used to describe the geometry of the space. Intuitively, we would say that that a symmetry of a space is a transformation under which the geometry of the space remains unchanged. Such a transformation is called an isometry, which can be discrete (e.g. a reflection), or continuous. The continuous isometries have associated vector fields, called Killing fields, named after the German mathematician Wilhelm Killing (1847-1923). These fields satisfy the Killing equation, which is developed in Chapter 4. Finding the symmetry properties of a manifold is a non-trivial task. Armed with the knowledge that isometries are related to the solutions of the Killing equation, we can attempt to solve it. However the Killing equation is an overdetermined system of partial differential equations, and subsequently may not have any solutions, let alone ones that are easy to find. The method of prolongation gives a system of first order differential equations, which is, in principle at least, solvable. It proceeds by adding dependent variables into the system to obtain a closed system. This process puts an immediate upper bound on the dimension of the solution space of the system, and thus tells you when to stop looking for solutions. It also produces candidate solutions that can be checked by substituting them back into the original equation. The prolongation method is described for the Killing equation in Chapter 5, where we look at the specific case of the 2-sphere. Symmetry is important in physics, where it is used to describe invariance such as conservation of momentum and energy. It is crucial to the study of Einstein's theory of General Relativity, in which exact solutions for the structure of space-time are near impossible to find without some symmetry assumptions. In the final chapter, we will look briefly at the structure of space-time as a manifold, and the symmetries of some of the space-time metrics.