Manifolds

Here, you find my whole video series about Manifolds in the correct order and I also help you with some text around the videos. If you want to test your knowledge, please use the quizzes that will come eventually, and consult the PDF version of the video if needed. When you have any questions, you can use the comments below and ask anything. However, without further ado let’s start:

Introduction

Manifolds is a video series I started for everyone who is interested in calculus on generalised surfaces one usually calls manifolds when some rules are satisfied. Some basic facts from my Real Analysis course and from my Functional Analysis course are helpful but I try to be as self-contained as possible. Let us start with the introduction and the definition of a topology.


PDF Version Quiz


With this you now know the foundations that we will need to start with this series. We will always work with topologies. So let us define some more notions.

Let’s get started

The notion of a interior point is something that comes immediately out when you look at the definition of a topology, which fixes open sets. More notions like closure and boundary of a set will also be explained now:


Watch Manifolds Part 2 PDF Version Quiz


Like in metric spaces, convergence is very important topic for a lot of calculations, like limits, derivatives and so on. It turn out that we need so-called Hausdorff spaces to get similar results:


Watch Manifolds Part 3 PDF Version Quiz


One important tool to construct new topological spaces is given by equivalence relations. This leads to a so-called quotient topology:


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Let us talk more about projective space which is defined by a quotient space:


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Now we introduce the concept of second-countable spaces which we need later to define manifolds. For this reason, we first need to define the notion of a base or basis of a topology.


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The next notion describes one of the most important concepts in topology: continuous maps. They are important because the conserve the whole structure of a topological space. Therefore, for invertible maps, we introduce the natural definition of a homeomorphism.


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A concept we will also need in the series about manifolds is known as compact sets:


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Finally, we can talk about the definition of a manifold:


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Let’s look at examples for manifolds.


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Another example for an abstract manifold is given by the projective space.


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Now we are ready to include more structure on our topological manifolds. These will be so-called smooth structures and they will make the manifold to a smooth manifold. Indeed, these will be the objects we want to study because there we can do calculus.


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Let us look at examples for such smooth structures and smooth manifolds. A lot of manifolds we discussed before already carry a smooth structure.


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Next, we discuss the notion of submanifolds, which is just a manifold found inside a larger one. Especially submanifolds of $ \mathbb{R}^n $ will be important later.


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