Introduction to Topology and Manifolds

These are notes I took on the first five lectures of the 2015 WE-Heraeus Winter School on Gravity and Light, held at Johannes Kepler University Linz to celebrate the 100th anniversary of Einstein’s theory of general relativity. All 24 lectures are available on YouTube. The later lectures dive into the physics of general relativity, but the first eight or so lectures are almost entirely mathematical, with the purpose of giving the background in topology and differential geometry needed to understand general relativity.

In my opinion, this course is one of the best introductions to topology and differential geometry available on YouTube. The lecturer, Frederic P. Schuller, is a marvel. (He even has a fan club on Facebook!) Then, without further ado, the notes:

Lecture 1: Topology

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Topology: Let \(M\) be a set. A topology \(\mathcal{O}\) is a subset \(\mathcal{O} \subseteq P(M)\) (i.e., a subset of the power set of \(M\)) such that

1. \(\emptyset \in \mathcal{O}, M \in \mathcal{O}\),
2. \(U \in \mathcal{O}, V \in \mathcal{O} \implies U \cap V \in \mathcal{O}\),
3. \(U_{\alpha} \in \mathcal{O} \implies \bigcup_{\alpha \in A} U_\alpha \in \mathcal{O}\), where \(A\) is some index set. This is just saying that it is closed under (possibly uncountable) union.

Standard topology: The topology defined on \(\mathbb{R}^d\) that you are used to. Open sets are defined as they are in analysis.

Topological space: A pair \((M, \mathcal{O})\) where \(M\) is some set and \(\mathcal{O}\) is a topology on that set.

Open set: If \(U \in \mathcal{O}\), we say that it is open.

Closed set: If \(M \setminus A \in \mathcal{O}\), we say that \(A\) is closed.

Continuity: Let \((M, \mathcal{O}_M)\) and \((N, \mathcal{O}_N)\) be topological spaces. Then a map \(f : M \to N\) is called continuous (with respect to \(\mathcal{O}_M\) and \(\mathcal{O}_N\)) if for all \(V \in \mathcal{O}_N\), we have that \(preim_f(V) \in \mathcal{O}_M\). I.e., continuous maps pull back open sets to open sets.

Composition of continuous maps: The composition of continuous maps is continuous.

Subset topology: Suppose \((M, \mathcal{O}_M)\) is a topological space and \(S \subseteq M\). We define \(\mathcal{O} \vert_S \subseteq P(S)\) as \(\mathcal{O} \vert_S := \{ U \cap S \mid U \in \mathcal{O}_M \}\). Called the “subset topology inherited from the topology on the superset,” or just the subset topology.

Restriction of a continuous map to a subset of the domain is continuous under the subset topology: Let \(f : M \to N\) be continuous, where you have already defined some \(\mathcal{O}_M, \mathcal{O}_N\). Let \(S \subset M\), and let \(f \vert_S : S \to N\) denote the restriction of \(f\) to \(S\). Note that whether \(f \vert_S\) is continuous depends on the topology chosen for \(S\). This theorem says that if you choose the subset topology for \(S\), then \(f \vert_S\) is indeed continuous.

Lecture 2: Topological Manifolds

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Manifold: A topological space \((M, \mathcal{O})\) is called a \(d\)-dimensional topological manifold if for all \(p \in M\), there exists some \(U \in \mathcal{O}\) with \(p \in U\) (an open set containing \(p\)), such that there exists a map \(x : U \to x(U) \subseteq \mathbb{R}^d\) (\(x\) maps \(U\) to some subset of \(\mathbb{R}^d\); here \(x(U)\) is the image of \(U\) under \(x\)) such that

1. \(x\) is invertible (\(x^{-1}: x(U) \to U\)).
2. \(x\) is continuous with respect to the subset topology \(\mathcal{O} \vert_U\) on \(U\) and the standard topology on \(\mathbb{R}^d\). (Or I guess, really the standard topology restricted to \(x(U)\).)
3. \(x^{-1}\) is also continuous.

These three conditions are equivalent to saying that \(x\) is a homeomorphism.

Note: This definition implies that \(x(U)\) is open. This is because \(x^{-1}\) is continuous so it must pull back open sets to open sets. We know that \(U\) is open. The preimage of \(U\) under \(x^{-1}\) is \(x(U)\), so \(x(U)\) must be open.

Chart: Borrowing terminology from the definition of a manifold above, a chart of the manifold \((M, \mathcal{O})\) is just a pair \((U, x)\), i.e., and open set and the homeomorphism that takes it to some subset of \(\mathbb{R}^d\). Then, the definition of a manifold is equivalent to saying that every point in your topological space has a chart (mapping to the same dimension).

Atlas: Intuitively a collection of charts that covers the manifold. Formally, it is a set

where each \((U_\alpha, x_\alpha)\) is a chart, \(A\) is some index set, and it must be true that \(M = \bigcup_{\alpha \in A} U_\alpha\).

Chart map: The map \(x : U \to x(U) \subseteq \mathbb{R}^d\) in some chart.

Coordinate maps: Consider a chart map \(x\). It is just some homeomorphism \(x : U \to x(U) \subseteq \mathbb{R}^d\). We can break it into its coordinate functions as follows: Let \(p\) be an arbitrary point in \(U\). Then define coordinate functions \(x^1, \dots, x^d : U \to \mathbb{R}\) such that \(x(p) = (x^1(p), \dots, x^d(p))\). (The coordinate maps are defined so this is true for any \(p\).) We call these \(x^i\)’s the coordinate maps.

(Note: The \(1\) is \(x^1\) is not an exponent. Treat it like a subscript; I’m just following his notation.)

Let \(p \in U\). Then \(x^1(p)\) is the first coordinate of the point \(p\) with respect to the chosen chart. \(x^2(p)\) is the second coordinate of \(p\), etc.

Chart transition map: Before explaining, here are two suggestive images:

The idea is that you have two charts \((U, x), (V, y)\) which cover an overlapping region in the domain, this being \(U \cap V\). You want a map from \(x(U \cap V)\) to \(y(U \cap V)\), where, e.g., \(x(U \cap V)\) denote the image of \(U \cap V\) under \(x\). Note that both \(x(U \cap V), y(U \cap V)\) are open subsets of \(\mathbb{R}^d\). Well, such a map is \(y \circ x^{-1}\), and it is in fact continuous since, as shown in the first lecture, the composition of continuous maps is continuous.

At an intuitive level, you can think about this as a scenario where you want to change your coordinate system, maybe from the Cartesian system to polar coordinates. Then the Chart transition map tells you how to do this. Note that both the Cartesian coordinate system and the polar coordinate system don’t really exist, in some sense. They are just being used to “chart out,” the real world, that being the space that contains \(U\) and \(V\).

Informally, the chart transition maps contain the instructions which show how to glue together the charts of an atlas.

Manifold philosophy: This isn’t a formal term, but the discussion in this section is so beautiful that I have to include it here. Often it is desirable or necessary to define the properties of a real-world object, say a curve \(\gamma : \mathbb{R} \to M\), where \(M\) is a set denoting the real-world, whatever that may mean, by judging suitable conditions not on the real-world object itself but on a chart-representation of the real-world object.

The advantage of this is that a chart map takes you into \(\mathbb{R}^d\), where you can make sense of things like continuity. (E.g., is \(\gamma\) continuous?) Here is a picture:

There is a danger though: properties given or defined this way may be ill-defined. This is because you chose the chart map \(x\). If you had chosen a different chart map, \(y\), then maybe the resulting function \(y \circ \gamma : \mathbb{R} \to \mathbb{R}^d\) isn’t continuous, even though \(x \circ \gamma : \mathbb{R} \to \mathbb{R}^d\) is.

In fact, this can’t happen. Observe:

The image in the top left shows the choice of two different chart maps \(x, y\). We are given that \(x \circ \gamma\) is continuous. But then it follows that \(y \circ \gamma\) is continuous because we can write \(y \circ \gamma\) in a different way by instead following the diagram downwards to \(x(U)\) and then straight up through \(U\) and finally to \(y(U)\). The resulting composition of functions is continuous since all the individual functions are continuous. (Note the use of the chart transition map \(y \circ x^{-1}\)!) So continuity is well-defined in this sense.

What about differentiability? Well, there we have a problem. \(x \circ \gamma\) might be differentiable, but following the path the long way around doesn’t help us because postcomposing a differentiable map (\(x \circ \gamma\)) by a continuous map (\(y \circ x^{-1}\)) does not necessarily preserve differentiability. (The postcomposition can introduces “notches.”) So we have to be more careful when talking about differentiability. (See Lecture 4.)

Maximal atlas: The atlas that contains all charts, or equivalently, the atlas that is not contained in any other atlas. It can be shown by Zorn’s lemma that a maximal atlas always exists and is unique.

Lecture 3: Multilinear Algebra

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Multilinear algebra is the study of vector spaces. So why are we interested in it if we care about manifolds? Well, that is because of the notion of a tangent space to a manifold, which will be defined later. Though manifolds need not carry any vector space structure, tangent spaces do, hence this lecture.

Vector space: A vector space is a triple \((V, +, \cdot)\) where

1. \(V\) is a set,
2. \(+ : V \times V \to V\) is an “addition” map
3. \(\cdot : \mathbb{R} \times V \to V\) is “scalar multiplication” or “S-multiplication,”

satisfying “CANIADDU”:

C^+ommutative law: \(v + w = w + v\) (true for all \(v, w \in V\))

A^+ssociative law: \((u + v) + w = u + (v + w)\)

N^+eutral element: \(\exists O : \forall v \in V : V + O = V\)

I^+nverse elements: \(\forall v \in V : \exists (-v) \in V : v + (-v) = O\)

Associative: \(\lambda \cdot (\mu \cdot V) = (\lambda \cdot \mu) \cdot V\) (\(\forall \lambda, \mu \in \mathbb{R}\))

Distributive: \((\lambda + \mu) \cdot v = \lambda \cdot v + \mu \cdot v\)

Distributive: \(\lambda \cdot v + \lambda \cdot w = \lambda \cdot (v + w)\)

Unitary law: \(1 \cdot v = v\) (where \(1\) is \(1\) from the real numbers)

Here ^+ means it has to do with addition—true for the first four. The latter four connect addition and scalar multiplication.

Note: You have to be careful because there are two kinds of multiplication and addition happening above. E.g., in the associative law in the fifth line, the \(+\) in \(\lambda + \mu\) is not the \(+\) in \((V, +, \cdot)\); rather, it is just addition on the real numbers. So be sure to differentiate between addition/multiplication as part of the vector space and the addition/multiplication of real numbers, since the same symbols are used for both.

Vector: An element of a vector space is often referred to, informally, as a vector.

Linear maps: These are the structure-respecting maps between vector spaces. Formally, let \((V, +_V, \cdot_V)\) and \((W, +_W, \cdot_W)\) be vector spaces. Then a map \(\phi: V \to W\) is linear if

1. \(\phi(v +_V \widetilde{v}) = \phi(v) +_W \phi(\widetilde{v})\),
2. \(\phi(\lambda \cdot_V v) = \lambda \cdot_W \phi(v)\).

Note that you need to have these vector spaces defined to even talk about linear maps!

Notation: If \(\phi: V \to W\) is linear, we may express this as \(\phi : V \tilde{\to} W\).

Composition of linear maps is linear: Pretty self-explanatory. In the diagram below, \(\phi \circ \psi\) is indeed linear.

Vector space of homomorphisms: Let \((V, +_V, \cdot_W)\) and \((W, +_W, \cdot_W)\) be vector spaces. Define \(\text{Hom}(V, W)\) to be the set of all linear maps from \(V\) to \(W\), i.e., \(\text{Hom}(V, W) := \{ \phi : V \tilde{\to} W \}\). Note that so far this is just a set, but we can make it a vector space by defining addition and scalar multiplication. We will use circles to distinguish these:

1. \(+ : \text{Hom}(V, W) \times \text{Hom}(V, W) \to \text{Hom}(V, W)\) is defined as \((\phi, \psi) \mapsto \phi \oplus \psi\) where \((\phi \oplus \psi)(v) := \phi(v) +_W \psi(v)\).
2. \(\odot\) is defined analogously.

\((\text{Hom}(V, W), \oplus, \odot)\) is the vector space of homomorphisms.

(A homomorphism is the generalized version of a linear map, where you needn’t be working in vector spaces. See this for the distinction.)

Dual vector space: A heavily-used special case of what we just did. Let \((V, +, \cdot)\) be a vector space. Then the dual vector space \(V^*\) is defined as the set of all linear maps from \(V\) to \(\mathbb{R}\). Note that this definition only makes sense if \(\mathbb{R}\) is also a vector space, and indeed it is—with normal multiplication and addition. Formally,

and we see it is just one possible (very simple) vector space of homomorphisms!

Well, so far \(V^*\) is just a set. But \((V^*, \oplus, \odot)\) is a vector space, where we define \(\oplus, \odot\) as we would for \(\text{Hom}(V, \mathbb{R})\) in the definition for “vector space of homomorphisms” above. \((V^*, \oplus, \odot)\) is called the dual vector space (to \(V\)).

Covector: \(\phi \in V^*\) is called, informally, a covector. (A covector is also a vector in its own right of course.)

Tensor: Let \((V, +, \cdot)\) be a vector space. An \((r, s)\)-tensor \(T\) over \(V\) is a multilinear map

I.e., it eats \(r\) covectors and \(s\) vectors. Multilinear means linear in each coordinate. In the line above, we abuse notation and also use \(\tilde{\to}\) to denote multilinear.

Example of a \((0, 2)\)-tensor: Define \(P\) to be the vector space of polynomials of degree up to, say, \(5\). Define \(g : P \times P \tilde{\to} \mathbb{R}\) as

This is a \((0, 2)\)-tensor over \(P\).

An inner product is an example of a \((0, 2)\)-tensor. Linear maps are isomorphic to \((1, 1)\)-tensors (source). Actually, there is a really good explanation of this latter fact at the end of the tutorial for this lecture. Fast forward to 48:33.

The reason \(V\) and \(V^*\) are so important is that all tensors are constructed from elements of \(V\) and \(V^*\), so they are the building blocks, so to speak.

Note that in this discussion of tensors, we did not talk about coordinates or bases! This is very important.

Vectors and covectors as tensors: Recall: An element \(\phi \in V^*\) is a linear map \(\phi : V \tilde{\mapsto} \mathbb{R}\). This means that \(\phi\) is a \((0, 1)\)-tensor. Thus, any covector is a \((0, 1)\)-tensor.

Let \(v \in V\). You can show that if \(\dim V < \infty\), then \(V = (V^*)^*\). This means, by the definition of the dual space, that \(v\) is also a linear map \(v : V^* \tilde{\to} \mathbb{R}\), which in turn means that \(v\) is a \((1, 0)\)-tensor. Thus, any vector is a \((1, 0)\)-tensor.

Bases: Let \((V, +, \cdot)\) be a vector space. A subset \(B \subset V\) is called a basis if for all \(v \in V\) there exists a unique finite subset \(F \subset B\) (let \(F = (f_1, \dots, f_n)\)) such that there exist unique numbers \(v^1, v^2, \dots, v^n\) such that \(v = v^1 f_1 + \cdots + v^n f_n\).

Dimension: If there exists a basis \(B\) with finitely many elements, say \(d\) many, then we define \(\dim V := d\). (You can show that this doesn’t depend on the basis you pick, so it is well-defined.)

Vector components with respect to a basis: Let \((V, +, \cdot)\) be a finite-dimensional vector space. Having chosen a basis \(e_1, \dots, e_n\) of \((V, +, \cdot)\), we may uniquely associate \(v \mapsto (v^1, \dots, v^n)\). These are called the components of \(v\) with respect to the chosen basis.

The dual basis: Having chosen a basis \(e_1, \dots, e_n\) for \(V\), we could choose a basis \(\epsilon^1, \dots, \epsilon^n\) for \(V^*\) that has nothing to do with our basis for \(V\). But it is more economical to pick the \(\epsilon^i\)’s as follows: Pick \(\epsilon^i\) such that

This defines a unique basis of the dual space. Thus, we call it the dual basis.

Components of tensors: Let \(T\) be an \((r, s)\)-tensor over a finite-dimensional vector space \(V\). Let \(e_1, \dots e_n\) be a basis of \(V\) and let \(\epsilon^1, \dots, \epsilon^n\) be “the dual basis” (see definition above) of \(V^*\). Then define the \((r + s)^{\dim V}\) many real numbers

where \(i_1, \dots, i_r, j_1, \dots, j_s\) vary over all elements of \(\{ 1, \dots, \dim V \}\) (repeats allowed). We call these numbers the components of the tensor \(T\) with respect to the chosen basis.

Why is this useful? Well, knowing the components of a tensor along with the corresponding basis allows you to completely reconstruct the tensor. This is because you are able to see how the tensor acts on any input given this information.

He shows an example with this using a \((1, 1)\)-tensor:

The point is that you can express any input in the basis and dual basis and then use multilinearity to pull the sum out. The tensor components give you how the tensor acts on the corresponding expressions with only basis and dual basis vectors as the inputs.

He also introduces Einstein summation notation at the bottom. Notice that he uses superscripts for the dual basis vectors as well as the coefficients of the dual basis vectors. This is on purpose, as it allows one to use the special notation.

The multilinearity of tensors is key to making Einstein notation work since it means that it doesn’t matter whether the summation is inside or outside of the tensor.

Lecture 4: Differentiable Manifolds

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Let \((M, \mathcal{O})\) be a topological manifold. In this lecture, we want to define a notion of differentiability for

1. curves \(\mathbb{R} \to M\),
2. functions \(M \to \mathbb{R}\),
3. and maps \(M \to N\) where \(N\) is some other manifold.

First we focus on a curve \(\gamma : \mathbb{R} \to M\).

We already know at this point what it means for a curve \(\gamma\) on a manifold to be continuous. Well, at least we know what continuity means if the curve fits in a single chart \((U, x)\). Then we can look at the function \(x \circ \gamma: \mathbb{R} \to \mathbb{R}^d\), and if it is continuous, then we can say that \(\gamma\) is continuous. Recall that the choice of the chart map \(x\) doesn’t matter when it comes to continuity—see the Manifold philosophy section under the Lecture 2: Topological Manifolds notes. But as discussed in the same section, with differentiability, the choice of the chart map does matter, so we have to be careful.

Here is an image of the problem:

(The line that got cut off at the bottom is “may not be undergraduate differentiable.”)

The idea is to reason about the differentiability of \(\gamma\) by first reasoning about the differentiability of pieces of it using charts, and then combine the pieces in some way to get a global picture. In this case, we have two charts \((U, x), (V, y)\) where both \(U\) and \(V\) contain some portion of \(\gamma\). Well, the portion they both contain is the only portion where a conflict could occur (one chart says \(\gamma\) is differentiable and the other doesn’t), so it is \(U \cap V\) that we focus on. Suppose that \(x \circ \gamma : \mathbb{R} \to \mathbb{R}^d\) is undergraduate-differentiable (basic real analysis definition of differentiability). Is \(y \circ \gamma : \mathbb{R} \to \mathbb{R}^d\) also necessarily undergraduate-differentiable? We can express \(y \circ \gamma\) differently as

where recall that \(y \circ x^{-1}\) is the chart transition map, which is always continuous. So we are postcomposing a differentiable function \((x \circ \gamma)\) with a continuous one \((y \circ x^{-1})\). This need not be differentiable because the continuous function could introduce kinks. Thus, this approach simply does not work.

We can create similar charts for functions \(M \to \mathbb{R}\) and maps \(M \to N\). If you do this, you will end up with the same problem.

What is the remedy?

Compatible charts: Above, we used any imaginable charts on the topological manifold \((M, \mathcal{O})\). To emphasize this, we may say that we took \(U\) and \(V\) from the maximal atlas \(\mathcal{A}\) of \((M, \mathcal{O})\). (See the definition here.)

What if we are careful and only keep a subset of charts (still an atlas) such that all transition functions are differentiable? Then everything would work out.

Definition: Two charts \((U, x)\) and \((V, y)\) of a topological manifold are called “flower”-compatible if either

• \(U \cap V = \emptyset\), or
• if \(U \cap V \ne \emptyset\), then the chart transition maps \(y \circ x^{-1} : x(U \cap V) \mapsto y(U \cap V)\) and \(x \circ y^{-1} : y (U \cap V) \to x(U \cap V)\) (note that they are inverses of each other and both go from \(\mathbb{R}^d\) to \(\mathbb{R}^d\)) have the undergraduate version of the flower property. (It is necessary to check both maps; one can have the property, e.g., being \(C^\infty\), while the other doesn’t. See the tutorial for an example of this.)

So in this case, we replace “flower” with differentiable to get what we need. (“Flower” is just a placeholder for a specific property, such as being twice continuously differentiable. See the table below for some different properties you could replace “flower” with.)

Flower-compatible atlas: An atlas \(\mathcal{A}_{flower}\) is a flower-compatible atlas if any two charts in \(\mathcal{A}_{flower}\) are flower-compatible.

Flower-manifold: A flower-manifold is a triple \((M, \mathcal{O}, \mathcal{A}_{flower})\), where the first pair is a topological manifold and the last entry is a flower-compatible atlas of the manifold.

Options for flower:

Why do we care about the derivative being continuous? I.e., why not define \(C^1\) as just maps that are differentiable once, whether or not that derivative is continuous. Well, recall that there are functions which are differentiable but do not have continuous partial derivatives (example). It would be very nice to have continuous partial derivatives, and the derivative being continuous is exactly what we need for this to happen; see Theorem 9.21 in Rudin:

Summary: The more properties you want your manifold to satisfy, the more restrictive you need to be when picking your charts.

I want to emphasize something here. Suppose that you want to check whether a curve \(\gamma : \mathbb{R} \to M\) is differentiable on a manifold \(M\). The point is that, you need at least for the manifold to be \(D^1\) to even talk about the differentiability of \(\gamma\); otherwise, you run into the problem discussed in the beginning. However, it is totally fine if your transition functions are even more than differentiable, e.g., they could be \(5\)-times continuously differentiable. The point though is that the transition functions need to at least satisfy the property you want to show that \(\gamma\) satisfies. You can have a curve \(\gamma\) that is only \(C^3\) (but not \(C^4\)) on a manifold \(M\) if \(M\) is, say, \(C^7\). However, you can’t have a \(C^7\)-curve on a \(C^3\)-manifold.

\(C^k\)-atlases contain \(C^\infty\)-atlases: It can be shown that for \(k \ge 1\), any \(C^{k}\) atlas \(\mathcal{A}_{C^k}\) of a topological manifold contains as a sub-atlas a \(C^\infty\)-atlas.

This theorem implies that the truly difficult part is going from a \(C^0\) atlas to a \(C^1\) atlas, since once we get a \(C^1\) atlas, we know there is some way to keep removing charts so as to end up \(\ell\)-times continuously differentiable for any \(\ell\) we want.

Smooth manifolds: Due to the theorem above, we may always consider \(C^\infty\)-manifolds, called smooth manifolds, unless we wish to define Taylor expandability or complex differentiability. (By “consider” we mean that if you have a \(C^1\)-manifold, it can actually be made into a \(C^\infty\)-manifold.)

Rephrased: A smooth manifold is a triple \((M, \mathcal{O}, \mathcal{A})\) where the first two entries form a topological manifold and \(\mathcal{A}\) is a \(C^\infty\) atlas.

Isomorphic/isomorphism: Let \(\phi: M \to N\) be some map. If \(M, N\) are naked sets (sets with no additional structure), the structure-preserving maps are the bijections (invertible maps).

We say that two sets are set-theoretically isomorphic, denoted \(M \cong_{set} N\), if there exists a bijection \(\phi : M \to N\) between them. E.g., \(\mathbb{N} \cong_{set} \mathbb{Z}, \mathbb{N} \cong_{set} \mathbb{Q}\), etc.

Now consider sets equipped with topologies: \((M, \mathcal{O}_M), (N, \mathcal{O}_N)\). What does it mean to say that they are topologically isomorphic, denoted \((M, \mathcal{O}_M) \cong_{top} (N, \mathcal{O}_N)\)? “Topologically isomorphic” is equivalently termed “homeomorphic.” Well, we say this is the case if there exists a bijection \(\phi: M \to N\) such that \(\phi, \phi^{-1}\) are continuous. This is the structure-preserving map. Note that this is a stronger notion than set-theoretically isomorphic.

We can say two vector spaces are vector-space isomorphic, \((V, +_V, \cdot_V) \cong_{vec} (W, +_W, \cdot_W)\) if there exists a bijection \(\phi: V \to W\) such that \(\phi, \phi^{-1}\) are both linear.

Diffeomorphic/diffeomorphism: This is just the specific terminology we use for isomorphisms of smooth manifolds.

Two \(C^\infty\)-manifolds \((M, \mathcal{O}_M, \mathcal{A}_M), (N, \mathcal{O}_N, \mathcal{A}_N)\) are said to be diffeomorphic if there exists a bijection \(\phi : M \to N\) such that \(\phi, \phi^{-1}\) are both \(C^\infty\) maps.

What does it mean for \(\phi : M \to N\) to be \(C^\infty\)? Well, it needs to be \(C^\infty\) over any choice of charts in \(M, N\) when \(\phi\) is restricted to these portions. So pick such arbitrary charts \((U, x)\) in \(M\) and \((V, y)\) in \(N\), and consider:

So we actually look at the map \(y \circ \phi \circ x^{-1} : \mathbb{R}^d \to \mathbb{R}^d\) and check whether this is \(C^\infty\).

Of course, we need to check that with alternative charts, we still get \(C^\infty\)-ness:

This works out fine though because the transition functions are \(C^\infty\), since we are working in smooth manifolds.

How to think about diffeomorphic smooth manifolds: A sphere of any size is diffeomorphic to an ellipsoid of any size. A sphere is not diffeomorphic to a sphere where you have introduced (sharp) folds. It would be homeomorphic to it though. The point is that a diffeomorphism doesn’t say anything about shape; only that one of the objects doesn’t have folds.

Making \(C^\infty\)-manifolds out of \(C^0\)-manifolds: The number of \(C^\infty\)-manifolds you can possibly make out of a \(C^0\)-manifold (by cutting away charts I guess) up to diffeomorphism is given in the table below:

Note how many more there are for the dimension of spacetime (\(4\))!

Differentiability of curves, functions, and maps on manifolds: I realize that I perhaps did not answer the very first question posed in the clearest terms possible. Let’s start with a curve \(\gamma : \mathbb{R} \to M\). To even make sense of it being differentiable, we need \(M\) to be at least a \(D_1\)-manifold, or equivalently, come equipped with a \(D_1\)-comparable atlas. Otherwise, it doesn’t make sense to ask about differentiability due to the issue described above.

If this is true, then we can proceed. We say \(\gamma\) is differentiable if it is differentiable with respect to any chart in the \(D_1\)-comparable atlas. Specifically, fix some arbitrary chart \((U, x)\). Then we need \(x \circ \gamma : \mathbb{R} \to \mathbb{R}^d\) to be differentiable in the real analysis sense.

Now consider a function \(f : M \to \mathbb{R}\). We again need \(M\) to be at least a \(D_1\)-manifold. The condition is basically the same. Let \((U, x)\) be an arbitrary chart. Then we need \(f \circ x^{-1} : \mathbb{R^d} \to \mathbb{R}\) to be differentiable in the real analysis sense. There is a nice diagram of this from the tutorial:

(Their \(f\) goes from \(M\) to \(\mathbb{R}^d\) instead of \(\mathbb{R}\), but the idea is the same.)

One question I had was: Could we equivalently consider the differentiability of the inverse of \(f \circ x^{-1}\), that being \(f^{-1} \circ x : \mathbb{R} \to \mathbb{R}^d\)? Well, the inverse of a differentiable function is differentiable. However, the issue is that \(f\) need not be invertible, as far as I can tell. If \(f^{-1}\) doesn’t exist, then of course \(f^{-1} \circ x\) doesn’t make sense. Note that \(f \circ x^{-1}\) is always well-defined since \(x\) is invertible by the definition of a chart map.

As an aside, here is a nice diagram of the curve and function cases from the tutorial:

They call \(x \circ \gamma\) and \(f \circ x^{-1}\), the two functions which you check for undergraduate differentiability to determine if the underlying curve/function on the manifold is differentiable, the “representative of the function” (for \(f \circ x^{-1}\)) and the “representative of the curve” (for \(x \circ \gamma\)). Not sure whether this is standard terminology.

Finally, consider a map \(g : M \to N\). Suppose \(M\) is \(d_M\)-dimensional and \(N\) is \(d_N\)-dimensional. Again, let both manifolds be \(D^1\). Let \((U, x)\) be an arbitrary chart of \(M\) and let \((V, y)\) be an arbitrary chart of \(N\). Then I’m guessing that \(g\) is differentiable as long as \(y \circ g \circ x^{-1} : \mathbb{R}^{d_M} \mapsto \mathbb{R}^{d_N}\) is. (He didn’t go over this formally in lecture, but I feel like this has to be right.) Note that this is basically a combination of the rule for functions with the rule for curves. (Of course, “map” and “function” are interchangeable; he just used different terms to distinguish between the different possible domains/codomains.)

Lecture 5: Tangent Spaces

Link

Notation: \(+_\mathbb{R}, \cdot_\mathbb{R}\) denote addition and multiplication on the reals. \(\oplus, \odot\) represent addition and scalar multiplication defined for a vector space.

Start of the lecture:

Motivating question: What is the velocity of a curve \(\gamma\) at a point \(p\) on a manifold \(M\)?

Velocity: Let \((M, \mathcal{O}, \mathcal{A})\) be a smooth manifold and let the curve \(\gamma : \mathbb{R} \to M\) be at least \(C^1\). Suppose \(\gamma(\lambda_0) = p\) for some \(\lambda_0 \in \mathbb{R}\). The velocity of \(\gamma\) at \(p\) is a linear map

We need a vector space to talk about a linear map. Here \(C^\infty (M)\) is the set of smooth functions from \(M \to \mathbb{R}\), equipped with pointwise addition and multiplication as its vector-space operations:

equipped with the operations \((f \oplus g)(p) := f(p) +_\mathbb{R} g(p), (\lambda \odot g)(p) = \lambda \cdot_\mathbb{R} g(p)\).

The map \(v_{\gamma, p}\) is defined as

“It takes \(f\) and maps it to the derivative of \(f\) after \(\gamma\) evaluated at \(\lambda_0\).”

This is really a directional derivative. “Vectors in differential geometry survive as the directional derivative they induce.”

Tangent vector space: For each point \(p \in M\), we define the set \(T_p (M)\), called the tangent space to \(M\) at \(p\), as the collection of all possible tangent vectors to all possible smooth curves through the point. Formally,

Notice that the point \(p\) stays fixed, but we vary over all possible smooth curves \(\gamma\).

What do we mean by tangent vector? Well, as you can see from the formal definition, by tangent vector we really mean “velocity map.” However, as described in the definition for velocity, the velocity vectors form a vector space (hence tangent “vector”).

Note that we make no reference to an ambient space in this definition. He draws two pictures here to illustrate this:

The picture on the bottom left is the better way to think about it—we don’t bring in any ambient space. The picture on the right is actually fine as well—there are theorems showing that you can always embed a manifold into a suitable ambient space and then draw this picture, but he recommends that you don’t think about it this way.

Velocities at a point form a vector space: A key observation is that the velocities at a point form a tangent space. Formally, \(T_p M\) can be made into a vector space.

To do this, we need to define an addition and scalar multiplication:

1. We define addition \(\oplus : T_p M \times T_p M \to \text{Hom}(C^\infty (M), \mathbb{R})\) as \((v_{\gamma, p} \oplus v_{\delta, p})(f) := v_{\gamma, p}(f) +_\mathbb{R} v_{\delta, p})(f)\). (Here, \(f \in C^\infty (M)\).) This is just pointwise addition. Here \(\gamma, \delta\) are two curves; it makes sense that two different curves generate two different elements of \(T_p M\) per its definition. Of course, \(p\) must stay fixed.

2. We define scalar multiplication \(\odot : \mathbb{R} \to T_p M \to \text{Hom}(C^\infty (M), \mathbb{R})\) as \((\lambda \odot v_{\gamma, p})(f) := \lambda \cdot_\mathbb{R} v_{\gamma, p}(f)\). Again, this is really just pointwise multiplication.

But wait a second! It is clear that the two maps \(\oplus, \odot\) defined above map into \(\text{Hom}(C^\infty (M), \mathbb{R})\), but this is not good enough. For these to be valid addition and scalar multiplication operations for the vector space, we need them to map back into \(T_p M\). So we need to show that

1. There exists a curve \(\sigma\) such that \(v_{\gamma, p} \oplus v_{\delta, p} = v_{\sigma, p}\).

2. There exists a curve \(\tau\) such that \(\lambda \odot v_{\gamma, p} = v_{\tau, p}\).

Showing that the outputs of the two operations can be written in this way for some \(\sigma, \tau\) means that they are indeed in \(T_p M\). I’m not going to copy it down, but the proof of these two points starts at about the 18-minute mark.

I stopped taking notes on this lecture at about the 25-minute mark.