Hello and welcome to my ongoing video course about Manifolds, already consisting of 58 videos. This series explores important topics in a structured order and is nearly complete, with more videos to be added in the future. Along with the videos, you’ll find helpful text explanations. You can test your knowledge using the quizzes and refer to the PDF versions of the lessons as needed. If you have any questions, feel free to ask in the community forum. Let’s dive in!
Part 1 - Introduction and Topology
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.
Content of the video:
00:00 Introduction
00:20 Overview
02:24 Stoke’s theorem as the goal
02:56 Metric Spaces
04:56 Definition Topology
07:29 Simple examples of topological spaces
09:07 Credits
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.
Part 2 - Interior, Exterior, Boundary, Closure
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:
Part 3 - Hausdorff Spaces
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:
Content of the video:
00:00 Introduction
00:25 Convergence in metric spaces
03:07 Convergence in topological spaces
03:59 Definition: convergence and limit
05:20 Example of non-unique limit
07:34 Definition: Hausdorff space
08:54 Credits
Part 4 - Quotient Spaces
One important tool to construct new topological spaces is given by equivalence relations. This leads to a so-called quotient topology:
Part 5 - Projective Space
Let us talk more about projective space which is defined by a quotient space:
Part 6 - Second-Countable Space
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.
Part 7 - Continuity
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.
Part 8 - Compactness
A concept we will also need in the series about manifolds is known as compact sets:
Part 9 - Locally Euclidean Spaces
Finally, we can talk about the definition of a manifold. We combine three properties and the last property will be called locally Euclidean, which means that a manifold can be flattened at least locally. We will use charts to do that. A whole collection of charts that cover the whole manifold is called an atlas.
Part 10 - Examples for Manifolds
Let’s look at examples for manifolds.
Part 11 - Projective Space is a Manifold
Another example for an abstract manifold is given by the projective space.
Part 12 - Smooth Structures
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.
Part 13 - Examples of Smooth Manifolds
Let us look at examples for such smooth structures and smooth manifolds. A lot of manifolds we discussed before already carry a smooth structure.
Part 14 - Submanifolds
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.
Part 15 - Regular Value Theorem in $\mathbb{R}^n$
For manifolds in $\mathbb{R}^n$, we have a nice theorem. It’s possible to describe them as preimages of regular values. This is known as the regular value theorem.
Part 16 - Smooth Maps (Definition)
By lifting the notion of differentiability from $ \mathbb{R}^n $ to manifolds with the help of charts, we can define so-called smooth maps between two smooth manifolds.
Part 17 - Examples of Smooth Maps
After defining the concept of a smooth map, we can look at some examples.
Part 18 - Regular Value Theorem (abstract version)
We already discussed the regular value theorem for submanifolds in $ \mathbb{R}^n $. However, we can lift the whole theorem to the abstract level and see that it also holds for submanifolds in any manifold.
Part 19 - Tangent Space for Submanifolds
This is the point where we introduce the first version for a tangent space. It’s very demonstrative to define this notion for submanifolds first.
Part 20 - Tangent Curves
In the next video, we want to give an alternative definition for the tangent space. This one has the advantage that we can also generalise it for abstract manifolds, which are not given as subsets of $ \mathbb{R}^n $. This will be the notion we will use for the rest of the series. It will be needed to define the differential of a smooth map.
Part 21 - Tangent Space (Definition via tangent curves)
Now we can finally define the tangent space in the general context, which means for every smooth manifold. For this, we will take the knowledge from the tangent curves from the last video and try to describe the essence of tanget vectors such that they also make sense for abstract manifolds that are not submanifolds in $ \mathbb{R}^n $. This will be done with equivalent relations and equivalence classes.
Part 22 - Coordinate Basis
For understanding the definition of the tangent space, it is helpful to describe it with basis vectors. Since we have an isomorphism with local charts, we can check what happens for the canonical basis from $ \mathbb{R}^n $ under this map. What we get is a basis in $ T_p(M) $, which we denote by $ ( \partial_1, \ldots, \partial_n ) $.
Part 23 - Differential (Definition)
After all this work and discussions about tangent spaces, we are now ready to define the differential of a smooth map, denoted by $d f_p$. We also define a new manifold as the disjoint union of all tangent spaces and call it the tangent bundle, denoted by $ TM $.
Part 24 - Differential in Local Charts
Here we look at the relation of the differential and the common Jacobian.
Part 25 - Differential (Example)
In this video, we discuss the notion of directional derivative again and will look at some concrete example for a differential of a smooth map.
Part 26 - Ricci Calculus
Let’s talk a little bit about the Ricci calculus, also called tenso calculus. We just give a short introduction how to translate the objects we discuss before into this new language. One important ingredient is that one suppresses a lot of details and deals with superscripts and subscripts for variables. This leads us to contravariant and covariant vectors, where more details will be discussed in future videos.
Part 27 - Alternating k-forms
For defining integration on manifolds, we have to some groundwork first. This means we have to learn some multilinear algebra, in particular, so-called alternating multilinear maps. In short, we will just call them k-forms.
Part 28 - Wedge Product
After introducing k-forms, we can also define a multiplication for the space of alternating multilinear maps. This is related to the tensor-product but this special version is called wedge-product because it’s written as $ \alpha \wedge \beta $.
Part 29 - Differential Forms
Finally, we can give the explicit definition of a differential form on a manifold. We will use them later for integration on manifolds.
Content of the video:
00:00 Introduction
01:11 Definition of k-forms on a manifold
04:00 Correction: It should be $\omega(f(p))$
4:18 Basis elements of k-forms
4:18 Vector space basis for k-forms
7:30 Example for 2-forms
8:35 Conclusion: local representation
8:35 Definition: differential form
Part 30 - Examples of Differential Forms
In order to understand how differential forms, we should look at some simple and important examples. We don’t have to go into complicated manifolds because the euclidean space already offers some important differential forms. In particular, we will look at volume forms and polar coordinates.
Part 31 - Orientable Manifolds
Here, let’s talk about orientated vector spaces. This means we distinguish positively orientated bases and negatively orientated bases. By applying this to manifolds, we can define so-called orientable manifolds, which will be very important for integration theorems later. However, it turns out that not all manifolds are orientable. The best-known counterexample is the Möbius strip.
Part 32 - Alternative Definitions for Orientations
It turns out that one has more possibilities for defining orientations on manifolds. Here we will prove one equivalence and state another.
Part 33 - Riemannian Metrics
With this video we finally put some geometry to the manifolds. These is done by using the tangent spaces at each point an inner products in them. The collection of all these inner products is usually called a Riemannian metric, named after the German mathematician Bernhard Riemann. Moreover, a smooth manifold together with a Riemannian metric is called a Riemannian manifold. These manifolds allow for a lot of calculations, like distances, angles, areas, and so on. Therefore, there are many applications for these manifolds, which we will discuss later on.
Part 34 - Examples for Riemannian Manifolds
Let’s look at some examples to get an idea how Riemannian metrics work. Indeed, for submanifolds in $ \mathbb{R}^n $, we should have a standard Riemannian metric coming from the standard inner product. We should check if everything fits together with our intuition and knowledge for geometry.
Part 35 - Canonical Volume Forms
We already know that the orientable manifolds have non-trivial volum forms. More precisely, if the dimension of $ M $ is $ n $, the there is a non-zero $ \omega \in \Omega^n(M) $. It turn out that for Riemannian manifolds we can choose a so-called canonical volume form, which is normalized with respect to the Riemannian metric. This one is related to the determinant of Gram, also knows as the Gramian.
Part 36 - Examples for Canonical Volume Forms
After the precise definition of canonical volume forms, we are ready to consider some concrete examples. In fact, we already know it for the sphere $S^2$. Let’s also look at graph surfaces in $ \mathbb{R}^3 $, which can be described by functions $ f: \mathbb{R}^2 \rightarrow \mathbb{R} $.
Part 37 - Unit Normal Vector Field
Before dive into integration on manifolds, I want to point out that canonical volume forms for submanifolds can be very easy to understand if you have a so-called continuous unit normal vector field.
Content of the video:
00:00 Introduction
00:50 Example picture
02:20 Definition: normal vector field
04:54 Definition: continuous unit normal vector field
08:13 Connection between volume form and normal vector field for submanifolds
11:39 Correction: Note that the orientations have to fit together such that the determinant gives a positive result for positively orientated vectors.
12:52 Example: sphere in the three-dimensional space
Part 38 - Integration for Differential Forms
Now finally, let’s start the integration theory for manifolds. We require some knowledge about Riemann integrals or Lebesgue integrals, which you can find in the Real Analysis courseand the Measure Theory course, respectively. So let’s first get a visualization about that. The best way to do this is to see the function inside the integral as a density. This gives us the correct connection to the integral of a differential form.
Part 39 - Integration on a Chart (Definition)
Now that we know that differential forms are the correct objects in an integration, we can try to lift the results from $ \mathbb{R}^n $ to general manifolds. This can just be done with a parameterization $ \varphi : \widetilde{U} \rightarrow U $. Therefore, we start our definition for integration by considering a single chart $ (U,h) $ first. On this one, we could motivate a decomposition like one does it for the Riemann integral and we get an nice general formula out.
Part 40 - Integral Over A Chart Is Well-Defined
This video is a little bit technical because we have to show the integral $ \int \omega $ is well-defned, which means that is does not depend on the chosen chart. One can say that this is some sort of substitution rule for this abstract integral. Indeed, it turns out that the common change-of-variable formula is the key ingredient in this prooof.
Part 41 - Measurable Sets and Null Sets
At the moment, we can only calculate integrals that completely lie inside a single chart. However, for a lot of applications, this is already sufficient if we are able to exclude some sets that have a volume of zero. In the common matter of speaking, we call these sets null sets. So let’s look at an example and at the general definition.
Part 42 - Integrable Differential Forms
Let’s continue the discussion about the definition of the integration. Here we will finally define the integral over a whole manifold. For this we have to consider countably many charts to split the manifold up. This is always possible because a manifold is second-countable by definition which implies that there is a countable atlas $ (U_k, h_k)_{k \in \mathbb{N} } $.
Part 43 - Integral is Well-Defined
The definition of the last video had one little flaw: we don’t know if two different decompositions lead to the same integral in the end. The proof of this fact will be the content of this video. It’s not hard to show if you already know how to deal with integrals in $ \mathbb{R}^n $.
Part 44 - Change of Variables
We already talked about the change of variables formula for integrals in $\mathbb{R}^n$. Since the integration on manifolds is defined by the common integration in $\mathbb{R}^n$, we can also lift this change of variables formula. What we get is an ** abstract change of variables formula** for manifolds. It’s really easy to remember and we can also prove it.
Part 45 - Manifolds with Boundary
You might already know that Stokes’s Theorem says something about a boundary of a manifold. However, we never defined a notion of manifolds with boundary until now. It turns out that is a straightforward generalization of a smooth manifold.
Part 46 - Example of a Manifold with Boundary
After discussing the general definition, we should look at an example for a manifold with boundary. Most importantly, we want to see if we can cover the whole manifold with charts that map to the half-space.
Part 47 - Tangent Space and Orientation on the Boundary
As mentioned before, we can lift a lot of notions from the original manifold definition to new manifold definition that also includes boundaries. In fact, this is a nice natural generalization such that one does not have to say much about this. However, for the important term of tangent vectors, it turns out that, at boundary points, we can distiguish three kinds: inward pointing, outward pointing, and also tangent vectors on $\partial M$. This is what we will discuss in detail and we will also see that the common orientation on the boundary $\partial M$ is inherited by the original orientation on $M$.
Content of the video:
00:00 Introduction
00:48 Visualization of the tangent space
02:35 Definition of tangent space on the boundary
04:13 Equivalent curves
06:17 Outward-pointing and inward-pointing tangent vectors
06:57 Riemannian manifold with boundary
08:03 Outward-pointing unit normal vector field
08:03 Inherited orientation of the boundary
12:15 Correction: any outward-pointing tangent vector will fix the orientation on the boundary. So it’s not necessary to have a Riemannian metric.
12:55 Credits
Part 48 - Stokes’s Theorem as the Fundamental Theorem of Calculus
We are ready to take big conceptual strides toward understanding Stokes’s Theorem. It turns out that we can see the familiar Fundamental Theorem of Calculus as a special case of the general theorem. In order to that we first have to define an orientation on zero-dimensional manifolds as well.
Part 49 - Cartan Derivatives
If we want to formulate Stokes’s theorem for more than just 1-forms, like in the last video, we have to extend the meaning of the $d$-operation to every $k$-form. This extension is known as the Cartan derivative or exterior derivative on manifolds. It turns out that we can do this in a unique way to get a whole sequence $d^{(0)}$, $d^{(1)}$, and so on.
Part 50 - Example of Exterior Derivative
We’ve already discussed how the exterior derivative should act locally. Based on this definition, we will prove the properties like we want them to have: the product rule and the complex property $d \circ d = 0$. In addition to that, we can look at an explicit example $\omega \in \Omega^1(\mathbb{R}^3)$ and calculate $d\omega$. We will see that this is completely related to the classical rotor definition $\mathrm{curl}$ of a vector field in $\mathbb{R}^3$.
Part 51 - Naturality of Cartan Derivative
The exterior derivative is natural in the sense that it compatible with the pullback. It short we can say $d$ commutes with $ f^\ast$. This is what we can prove now.
Part 52 - Generalized Stokes Theorem
Now we are finally ready to formulate Stokes’s Theorem in the general version. It tells us that an integral over a manifold $M$ can be reduced to an integral over the boundary $\partial M$. The crucial ingredient is that we have a compactely supported differential form and the exterior derivative of it. In short: $ \int_M d \omega = \int_{\partial M} \omega$.
Part 53 - Proof of Stokes’s Theorem (Half-Space)
In order to prove the generalized Stokes theorem on manifolds, we should first look at the simplest case, which means the half-space $\mathbb{R}^n_{\leq}$. We will see that we only need Fubini’s Theorem there and the fundamental theorem of calculus.
Other videos related to this topic:
Part 54 - Proof of Stokes’s Theorem (In One Chart)
It’s quite straightforward to lift the proof from the last video to a general manifold. The only thing we have to require then is that everything can be done with just one chart $(U,h)$. So here we assume that $\mathrm{supp}(\omega) \subseteq U$.
Part 55 - Proof of Stokes’s Theorem (General Case)
This is the third an final part in the proof of Stokes’s Theorem. The idea was straightforward so far: start with the simplest case and generalize step by step. And now, we’ve reached the general case as formulated in the theorem. The idea is also clear: try to write $\omega = \omega_1 + \cdots + \omega_N$ as a finite sum of differential forms where each $\omega_j$ has compact support inside a single chart. This means we can just apply the result from part 54 $N$ times. The key idea for the decomposition of $\omega$ is the so-called partition of unity.
Part 56 - Gauss’s Integral Theorem
The generalized Stokes theorem is something that we can only understand on manifolds and for differential forms. However, there is also a classical formulation for vector fields in $\mathbb{R}^n$, which is particularily helpful in physical theories in three dimensions. This special version is known as the divergence theorem or Gauss’s theorem, named after Carl Friedrich Gauß. The theorem can be stated as relation of the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed. Let’s prove it by using Stokes’s theorem.
Part 57 - Example of Gauss’s Integral Theorem
After the long proof of Stokes’s Theorem, we can finally look at an application. Indeed, a lot of examples can be found in the three-dimensional space since they come directly from physics. We will see that Gauss’s integral theorem implies Gauss’s law of electrodynamics. This means that we will observe the flux of a vector field out of a closed surface, which is the boundary of a three-dimensional manifold in $\mathbb{R}^3$.
Part 58 - Stokes’s Integral Theorem (Classical Version)
After we have already discussed one special classical case of the generalized Stokes theorem, we can finally go the special case which actually carries the same name as the general theorem. We will call it Stokes’s integral theorem in $ \mathbb{R}^3 $ and it’s, indeed, an important result in classical vector analysis that has a lot of applications in physics and related fields. The core ingredient is the curl of a vector field $ v: \mathbb{R}^3 \rightarrow \mathbb{R}^3 $, which is connected to the exterior derivative of one-forms.
Other videos related to this topic:
Summary of the course Manifolds
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