From First Principles

Mathematics Without Hand Waving

Differential Geometry

 

Hello and welcome to my differential geometry page.

For many years I’ve always wanted to write a series of articles on differential geometry because in my opinion the subject is extraordinarily broad and requires the student to know quite a large volume of other mathematics before starting. I thought I had a fairly unique perspective as I was self taught on the subject and made many mistakes, wrong turns, and dead-ends along the way.

Although the material itself begins modestly, only requiring a very small number of assumptions, the variations that one can easily add become quite numerous and complex. By saying this, differential geometry is a very creative mathematical subject and by studying it you get to discover some pretty awesome things.

My interest in the subject really began to peak in 2005: my final undergraduate year. I was studying a bachelor of science and although I started out with a 50% split of physics and mathematics I ended the degree fully immersed in the math side of things. This lead to an additional honours year of pure math. My choices weren’t exactly varied as our math department focused heavily on analysis – something I wasn’t very good at. However (and this is what really convinced me to do an extra year) there was one professor who taught general relativity. A ha! Something I can visualise.

To a 3rd-year math undergrad general relativity was a behemoth of a subject. What on Earth is a tensor? What is meant by all those Christoffel symbols? Clifford algebras, quaternions, manifolds, metrics, connections, curvature the list just kept going and going. But it was the fact that a lot of the concepts and the things that they would eventually describe were geometrical in nature. There’s a lot of visualising circles, rotations, translations, permutation and combinations of objects, grouping objects based on their geometrical properties, moving things around, things that are flat, things that are curved, you get the idea.

I didn’t like analysis because at that point the lectures were all about weird limits and infinite sequences which I could never visualise. But with differential geometry I could visualise a tensor; as an arrow from a vector space and its dual to a single real number (so I used to draw them on paper as arrows from two boxes in to a line to represent the real numbers. You can visualise the commutator by drawing a little triangle and dragging a vector around it and then observing any rotation with the original. Manifolds were rubber sheets. The tangent bundle can be visualised as a 3D box sitting atop a curved manifold assigning a tangent vector to each and every point on the manifold. Even other geometrical properties could be equipped to a manifold, such as a metric which itself can be visualised as a little arrow drawn from two boxes representing a choice of two input tangent vectors from the tangent space in to a line representing the real numbers. I’ll get to all this visualisation in the proceeding articles.

To make a long story short, differential geometry was all about drawing diagrams of spaces as boxes, arrows between them as tensors, and curved sheets as manifolds. So this is where we will start, with the concept of a manifold (and how to draw it). We will learn how to draw one and what we can do with it by adding structure. We’ll do this a lot, simply because we can, but it makes life much more interesting when we do.

In this series of articles I will take you on my path-optimised journey through differential geometry. I’ll be keeping the math to a minimum, we won’t spend a lot of time on proofs, but the key will be on visualising things so there will be a lot of accompanying diagrams that I hope will be able to explain what it is that differential geometry does for us.

Table of Contents

  1. Preliminary Concepts – things we’ll need
  2. Introducing the Manifold – the rubber sheet
  3. Differentiable Manifolds – adding our first bit of structure
  4. The Tangent Bundle – where did that come from?
  5. The Dual Bundle – the dark side of the tangent bundle
  6. Tensors – the workhorse of manifolds
  7. The Algebra of Tensors – things we can do with tensors
  8. Curves, Flows and Brackets – gettin’ yourself around on manifolds
  9. Wedge, Forms and Integration – generalising basic geometrical concepts
  10. Riemannian Manifolds – the home of general relativity
  11. General Relativity – why the universe can be explained with geometry

 

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