Manifolds Tutorial

Charts and Atlases

My M.Sc. Project at IIT Madras deal with Manifolds and some numerical computations on them.

The initial part of my M.Sc. project introduced Manifolds in general. I’ve put that up on the web. So if you are interested in a quick and dirty introduction to Manifolds, here it is! There are lots of diagrams and figures to make the concepts easy. Its a fun and fast tutorial that can be completed in a few hours.

Here is the Preface

This HTML document aims to introduce Manifolds. It has been derived (mainly verbatim) from parts of my M.Sc. Physics final project. I used Latex2HTML to convert my latex to the HTML you see here.

Manifolds are fundamental structures in Differential Geometry. The study of Manifolds is useful in various branches of Theoretical Physics, especially High Energy Physics and General Relativity. For instance, Einstein’s theory of General Relativity conjectures that space-time forms a 4 dimensional pseudo-Riemannian Manifold. Superstring Theory explains the compactification of extra dimensions by using Calabi-Yau Manifolds.

Manifolds are abstract mathematical spaces that look locally like \small{\mathbb {R}^n} but may have a more complicated large scale structure. The surface of Earth is a simple example: At small distances it looks like the Euclidean \small{\mathbb {R}^2} but from far away it is \small{\mathbb {S}^2}, the two dimensional surface of a sphere. The behaviour at the small scale and large scale can be totally different. For instance, in \small{\mathbb {R}^2} parallel lines never meet while all lines eventually meet in \small{\mathbb {S}^2}. Because all Manifolds are locally like Euclidean Space we can develop common mathematical techniques to study extremely different kinds of spaces.

We can define increasingly complicated structures on Manifolds so that we may do Calculus on them or define concepts of distance and angles on them. We may also want to study Manifolds in terms of complex variables and perform Complex Calculus on them. In this report we look at the whole hierarchy of Manifolds. We start from Simple Manifolds and progress to Differentiable Manifolds, Riemannian Manifolds and lastly Complex Manifolds. Within Complex Manifolds we study Hermetian Manifolds and Kahler Manifolds. Orbifolds, another special kind of Manifold, are also introduced. All the related mathematics and concepts such as Vector Fields, Tangent Spaces, Metrics, Curvature, Parallel Transport and Connection are explained.

Still interested? Here is the HTML document