Charts and Atlases, Manifolds and Smooth Structures

1. Manifolds

We will introduce the machinery necessary for defining a smooth manifold.

1.1. Charts

1. Definition. Let ${X\subset M}$ be some set. An ${n}$-dimensional Chart consists of

1. an open subset ${U\subset\mathbb{R}^{n}}$
2. a map ${\varphi\colon U\to X}$

such that ${\varphi}$ is an appropriate isomorphism (for topological manifolds, it is a homeomorphism; smooth manifolds require a diffeomorphism; and so on).

2. Remark. We call ${\varphi\colon U\to X}$ a Parametrization of ${X}$, and ${\varphi^{-1}\colon X\to U}$ a Local System of Coordinates.

3. Remark. Since ${\varphi}$ is an isomorphism, the literature mixes up using ${U\to X}$ and ${X\to U}$. Milner uses ${\varphi\colon U\to X}$, but John Lee uses the opposite convention.

4. Definition. Let ${(U,\varphi)}$, ${(V,\psi)}$ be two charts. We say they are Compatible if

1. the set ${(\varphi^{-1}\circ\psi)(V)\subset U}$ is an open set;
2. the set ${(\psi^{-1}\circ\varphi)(U)\subset V}$ is an open set;
3. the map ${\psi^{-1}\circ\varphi\colon\varphi^{-1}(\psi(V))\to\psi^{-1}(\varphi(U))}$ is smooth; and
4. the map ${\varphi^{-1}\circ\psi\colon\psi^{-1}(\varphi(U))\to\varphi^{-1}(\psi(V))}$ is smooth.

In particular, the charts are compatible if ${\varphi(U)\cap\psi(V)=\emptyset}$ is disjoint.

5. Remark. We refer to the maps ${\psi^{-1}\circ\varphi}$ as Transition Functions. The condition of smooth is ${C^{\infty}(\mathbb{R}^{n})}$, but different manifolds have different conditions (we could have ${C^{k}}$ charts, or ${C^{0}}$ charts, or analytic ${C^{\omega}}$ charts, or...).

In the older literature (e.g., Kobayashi and Nomizu's Foundations of Differential Geometry), the collection of transition functions form a gadget called a Pseudogroup.

6. Remark. We abuse notation, and could be more explicit by writing $$ \psi^{-1}\circ\varphi\colon\varphi^{-1}(\varphi(U)\cap\psi(V))\to\psi^{-1}(\varphi(U)\cap\psi(V)) \tag{1}$$

▸ Exercise 1. Prove chart compatibility is an equivalence relation.

1.2. Atlases

7. Definition. Let ${M}$ be a set. An (${n}$-dimensional) Atlas consists of a collection ${\{(U_{\alpha},\varphi_{\alpha})\mid\alpha\in A\}}$ of ${n}$-dimensional charts on ${M}$ such that

1. Covers ${M}$: $\displaystyle{\bigcup_{\alpha\in A}\varphi_{\alpha}(U_{\alpha})=M}$
2. Pairwise compatible: for any ${\alpha}$, ${\beta\in A}$ the charts ${(U_{\alpha},\varphi_{\alpha})}$ and ${(U_{\beta},\varphi_{\beta})}$ are compatible.

8. Definition. Two ${n}$-dimensional atlases on ${M}$, ${\mathcal{A}}$ and ${\mathcal{B}}$, are called Equivalent if their union ${\mathcal{A}\cup\mathcal{B}}$ is also an atlas. That is to say, if any chart of ${\mathcal{A}}$ is compatible with any chart of ${\mathcal{B}}$.

9. Remark. Remember: charts are compatible, but atlases are equivalent.

10. Lemma. Let ${\mathcal{B}}$ be an atlas, let ${(U,\varphi)}$ and ${(V,\psi)}$ be two charts not contained in ${\mathcal{B}}$. If ${(U,\varphi)}$ is compatible with every chart of ${\mathcal{B}}$, and if ${(V,\psi)}$ is compatible with every chart of ${\mathcal{B}}$, then ${(U,\varphi)}$ is compatible with ${(V,\psi)}$.

11. Theorem. Equivalence of atlases is an equivalence relation.

Proof: Let ${\mathcal{A}}$, ${\mathcal{B}}$, ${\mathcal{C}}$ be arbitrary atlases on ${M}$.

1. Reflexivity: ${\mathcal{A}}$ is equivalent to itself, since by definition any pair of charts in ${\mathcal{A}}$ are compatible.
2. Symmetry: let ${\mathcal{A}}$ and ${\mathcal{B}}$ be equivalent atlases, then ${\mathcal{B}}$ and ${\mathcal{A}}$ are equivalent atlases.
3. Transitivity: this is the nontrivial part. Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be equivalent atlases, and ${\mathcal{B}}$ be equivalent to ${\mathcal{C}}$. Then transitivity follows by considering arbitrary charts ${(U,\varphi)\in\mathcal{A}}$ and ${(V,\psi)\in\mathcal{C}}$, then applying Lemma 10.

Thus "equivalence of atlases" forms an equivalence relation. ∎

12. Proposition. The collection of atlases on a given set ${M}$ is a set, not a proper class.

Proof: The class of atlases is a subcollection of $$ \mathcal{X}=\mathcal{P}\left(\bigcup_{U\in\mathcal{P}(\mathbb{R}^{n})}\mathop{\rm Hom}\nolimits(U,M)\right) \tag{2}$$ where ${\mathop{\rm Hom}\nolimits(U,\mathbb{R}^{n})}$ is the collection of (appropriately smooth, or continuous, or holomorphic, or...) functions from ${U}$ to ${M}$. By ZF axioms, ${\mathcal{X}}$ is a set. ∎

1.3. Manifolds

13. Definition. Let ${M}$ be a set, let ${\mathcal{A}}$ be an ${n}$-dimensional atlas on ${M}$. We call a subset ${B\subset M}$ Open (with respect to ${\mathcal{A}}$) if for any chart ${(U,\varphi)\in\mathcal{A}}$ the preimage ${\varphi^{-1}(B)}$ is open (in ${U}$, and thus open in ${\mathbb{R}^{n}}$). In particular, the images ${\varphi(U)}$ are open.

14. Theorem. If two atlases ${\mathcal{A}_{1}}$ and ${\mathcal{A}_{2}}$ on ${M}$ are equivalent, then a subset ${B\subset M}$ is open with respect to ${\mathcal{A}_{1}}$ if and only it is open with respect ${\mathcal{A}_{2}}$.

15. Remark. This theorem shows an equivalence class of atlases on ${M}$ makes ${M}$ a topological space. We may therefore meaningfully speak about topological properties of ${M}$ (like compactness, connectedness, and so forth).

16. Corollary. Let ${\mathcal{A}}$ be an ${n}$-dimensional atlas for ${M}$. Then the collection of open sets with respect to ${\mathcal{A}}$ form a topology on ${M}$.

17. Definition. Let ${M}$ be a fixed set. A ${n}$-Dimensional Differential Structure (or ${n}$-Dimensional Smooth Structure) on ${M}$ consists of an equivalence class ${\mathfrak{D}}$ of ${n}$-dimensional atlases on ${M}$ such that

1. Second-Countable: ${\mathfrak{D}}$ contains an at most countable atlas;
2. Hausdorff: for any distinct ${p,q\in M}$, there exists disjoint open neighborhoods ${U,V\subset M}$ such that ${p\in U}$ and ${q\in V}$.

18. Remark (Smooth Structure using a Maximal Atlas). Equivalence classes are awkward to work with, and so it is more popular to consider maximal atlases. An atlas ${\mathcal{A}}$ is maximal if it contains all charts compatible with every chart in ${\mathcal{A}}$. Given an equivalence class ${\mathfrak{A}}$ of atlases, we may obtain a maximal atlas by considering $$ \mathcal{A}_{\text{max}} = \bigcup_{\mathcal{A}\in\mathfrak{A}}\mathcal{A}. \tag{3}$$ This may be used instead of an equivalence class of atlases in defining a differential structure, provided the second-countable axiom is reworded as: ${\mathcal{A}_{\text{max}}}$ contains an at most countable subatlas.

19. Remark (Convenient Fiction). No one actually constructs either a maximal atlas or a differential structure. We typically construct a smooth atlas on ${M}$, then announce we are working with the differential structure containing our atlas. Thus maximal atlases and, to some degree, differential structures are a convenient fiction.

20. Definition. A (Smooth) ${n}$-Dimensional Manifold consists of a set ${M}$ equipped with an ${n}$-dimensional differential structure.

21. Puzzle. Is this definition correct? By this, I mean: is an "${n}$-Dimensional Differential Structure" actually structure (in the sense of "stuff, structure, and properties")?

Introducing Groups to Beginners

[This is an experiment to see if some software to translate LaTeX to html works.]

1. Introduction. We will do some group theory. Here "group" refers to a "group of symmetry transformations", and we should think of elements of the group as functions mapping an object to itself in some particularly symmetric way.

2. Definition. A Group consists of a set ${G}$ equipped with

1. a law of composition ${\circ\colon G\times G\to G}$,
2. an identity element ${e\in G}$, and
3. an inverse operator ${(-)^{-1}\colon G\to G}$

such that

1. Associativity: For any ${g_{1}}$, ${g_{2}}$, ${g_{3}\in G}$, ${(g_{1}\circ g_{2})\circ g_{3}=g_{1}\circ(g_{2}\circ g_{3})}$
2. Unit law: For any ${g\in G}$, ${g\circ e=e\circ g=g}$
3. Inverse law: For any ${g\in G}$, ${g^{-1}\circ g=g\circ g^{-1}=e}$.

3. Effective Thinking Principle: Create Examples. Whenever encountering a new definition, it's useful to construct examples. Plus, it's fun. Now let us consider a bunch of examples!

4. Example (Trivial). One strategy is to find the most boring example possible. We can't use ${G=\emptyset}$ since a group must contain at least one element: the identity element ${e\in G}$. Thus the next most boring candidate is the group containing only the identity element ${G=\{e\}}$. This is the Trivial Group.

5. Example (Dihedral). Consider the regular ${n}$-gon in the plane ${X\subset\mathbb{R}^{2}}$ with vertices located at ${(\cos(k2\pi/n), \sin(k2\pi/n))}$ for ${k=0,1,\dots,n-1}$. We also require ${n\geq3}$ to form a non-degenerate polygon (${n=2}$ is just a line segment, and ${n=1}$ is one dot).

We can rotate the polygon by multiples of ${2\pi/n}$ radians. There are several ways to visualize this, I suppose we could consider rotations of the plane by ${2\pi/n}$ radians: $$ r\colon\mathbb{R}^{2}\to\mathbb{R}^{2} \tag{1}$$ which acts like the linear transformation $$ r \begin{pmatrix} x\\ y \end{pmatrix} := \begin{pmatrix} \cos(2\pi/n) & -\sin(2\pi/n)\\ \sin(2\pi/n) & \cos(2\pi/n) \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix}. \tag{2}$$ We see that the image of our ${n}$-gon under this transformation ${r(X)=X}$ remains invariant.

The other transformation worth exploring is reflecting about the ${x}$-axis, ${s\colon\mathbb{R}^{2}\to\mathbb{R}^{2}}$ which may be defined by $$ s \begin{pmatrix} x\\ y \end{pmatrix} := \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix}. \tag{3}$$ This transformation also leaves our polygon invariant ${s(X)=X}$.

We can compose these two types of transformations. Observe that ${s\circ s=\mathrm{id}}$ and the ${n}$-fold composition ${r^{n}=r\circ\dots\circ r=\mathrm{id}}$ both yield the identity transformation ${\mathrm{id}(x)=x}$ for all ${x\in\mathbb{R}^{2}}$. Then we have ${2n}$ symmetry transformations: ${\mathrm{id}}$, ${r}$, ..., ${r^{n-1}}$; and ${s}$, ${s\circ r}$, ..., ${s\circ r^{n-1}}$. What about, say, ${r\circ s}$? We find ${s\circ r^{k}\circ s=r^{-k}}$, so ${r^{k}\circ s = s\circ r^{-k}}$. Thus it's contained in our list of symmetry transformations.

The symmetry group thus constructed is called the Dihedral Group. Geometers denote it by ${D_{n}}$, algebraists denote it by ${D_{2n}}$, and we denote it by ${D_{n}}$.

6. Example (Rotations of regular polygon). We can restrict our attention, working with the previous example further, to only rotations of the regular ${n}$-gon by multiples of ${2\pi/n}$ radians. We can describe this group as "generated by a single element", i.e., symmetries are of the form ${r^{k}}$ for ${k\in\mathbb{Z}}$. This is an example of a Cyclic Group. In particular, it is commutative: any symmetries ${r_{1}}$ and ${r_{2}}$ satisfy ${r_{1}\circ r_{2}=r_{2}\circ r_{1}}$. These are special situations, let us carve out space to define these concepts explicitly.

7. Definition. We call a group ${G}$ Abelian if it is commutative, i.e., for any transformations ${f}$, ${g\in G}$ we have ${f\circ g = g\circ f}$. In this case, we write ${f\circ g}$ as ${f+g}$, using the plus sign to stress commutativity.

8. Definition. We call a group ${G}$ Cyclic if there is at least one element ${g\in G}$ such that ${\{g^{n}\mid n\in\mathbb{Z}\}=G}$ the entire group consists of iterates of ${g}$ and ${g^{-1}}$.

9. Example (Number Systems). Another few examples the reader may know are the familiar number systems under addition: the integers ${\mathbb{Z}}$, the rational numbers ${\mathbb{Q}}$, the real numbers ${\mathbb{R}}$, and the complex numbers ${\mathbb{C}}$. They are commutative groups.

10. Example (Infinite dihedral). We can take the infinite limit of the dihedral group to get the infinite dihedral group ${D_{\infty}}$. We formally describe it as consisting of "rotations" ${r}$ and "reflections" ${s}$ such that

1. ${r^{m}\circ r^{n} = r^{m+n}}$ for any ${m}$, ${n\in\mathbb{Z}}$;
2. ${s\circ r^{m}\circ s = r^{-m}}$ for any ${m\in\mathbb{Z}}$;
3. ${s\circ s = e}$;
4. ${r^{n}\circ r^{-n} = r^{-n}\circ r^{n} = e}$ for any ${n\in\mathbb{Z}}$, in particular ${r^{0}=e}$.

In this sense, the "infinite limit" turns rotations into something like the integers.

11. Example (Circular dihedral). A more intuitive "infinite limit" of the dihedral group is the symmetries of the unit circle ${S^{1}}$ in the plane ${\mathbb{R}^{2}}$. These are anti-clockwise rotations and reflection about the ${x}$-axis, but rotations are parametrized by a real parameter (the "angle"): $$ r_{\theta}~``=\!\!\mbox{"} \begin{pmatrix}\cos(\theta) & -\sin(\theta)\\ \sin(\theta) & \cos(\theta) \end{pmatrix}. \tag{4}$$ Here we write an "equals" sign in quotes because this is the intuition. A group is abstract, whereas the matrix is a concrete realization of the symmetry.

The reader should verify the axioms for a group are satisfied, with the hint that ${r_{\theta}\circ r_{\phi} = r_{\theta+\phi}}$ and the usual relation between reflection and rotation holds.

This group is called the Orthogonal Group in 2-dimensions.

Exercises

Exercise 1. Is ${\mathbb{Z}}$ a cyclic group? Is ${\mathbb{C}}$ a cyclic group?

Exercise 2. Is the non-negative integers ${\mathbb{N}_{0}}$ a group under addition? Under multiplication?

Exercise 3. Are the positive real numbers ${\mathbb{R}_{\text{pos}}}$ a group under multiplication?

Exercise 4. Pick your favorite polyhedron in 3-dimensions. Determine its symmetry group.

Exercise 5. Complex conjugation acts on ${\mathbb{C}}$ by sending ${x+i\cdot y}$ to ${x-i\cdot y}$. Does this give us a symmetry group?

Exercise 6 (challenging). If we consider polynomials with coefficients in, say, rational numbers (denoted ${\mathbb{Q}[x]}$ for polynomials with the unknown ${x}$), then how can we form a symmetry group of ${\mathbb{Q}[x]}$?

Exercise 7 (General Linear Group). Take ${n\in\mathbb{N}}$ to be a fixed positive integer, preferably ${n\geq2}$. Consider the collection of invertible ${n}$-by-${n}$ matrices with entries which are rational numbers $${\mathrm{GL}(n, \mathbb{Q}) = \{ M\in\mathrm{Mat}(n\times n, \mathbb{Q}) \mid \det(M)\neq0\}.}$$ Prove this is a group under matrix multiplication.

TeX macro for normal operator ordering

I've always been bothered with normal operator ordering, writing $:O(a)O(b):$ always produces bad results.

The quick fix I've been using is the following:

\def\normOrd#1{\mathop{:}\nolimits\!#1\!\mathop{:}\nolimits}

%%
% example:
% \begin{equation}
% \normOrd{a(z)b(\omega)} = a(z)_{+}b(\omega)+(-1)^{\alpha\beta}b(\omega)a(z)_{-}
% \end{equation}
%%

Which in practice looks like:

How I got this solution

I determined this solution iteratively after many different attempts, which I shall enumerate along with the problems they each had.

However, using mere colons :a(z)b(\omega): = ... produces the following:

Being clever, I asked myself "Hey, why not write :x\colon for the normal ordering?" This was clever, but wrong. Consider the following example:

g = :x\colon

Producing:

Not one to give up easily, I found a \cocolon definition on tex.stackexchange. Trying that instead:

g = \cocolon x\colon = y

Produces strange extra whitespace on the right:

After examining the co-colon code, I just determined that something along the lines of

% rough draft definition #1
\def\normOrd#1{\mathrel{:}\!#1\!\mathrel{:}}

would work. This didn't quite work, the whitespacing was strange. So instead I just use \mathop{:}\nolimits..., which produces the desired result.

Revising my Notes on General Relativity

So I've been revising my notes on general relativity, and I've found several things worth mentioning.

1. Equivalence Principle. The equivalence principle gives us geometry. This is often poorly described (I too committed this error in my drafts).

The equivalence principle tells us neither the composition of a body nor its mass determines its trajectory in a gravitational field. So gravity determines paths, and this gives us geometry.

Moreover, there are different equivalence principles which should be mentioned. I yielded to this, and became incoherent (alas!). The trick is to stick this into a box, for the interested reader to find out more about it, but not obstruct the writing.

2. Coordinates for Black Hole. Different coordinates for the Schwarzschild solution are described beautifully in Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, and Eduard Herlt's Exact Solutions of Einstein’s Field Equations (Cambridge University Press, 2d edition, 2009).

3. Manifolds, Mathematics. I think I ought to examine Christopher Isham's Modern Differential Geometry for Physicists for a Physicist's differential geometry.

I should like to discuss the exponential map, which relates paths to geometry (as alluded in the equivalence principle discussion).

Most readers probably will agree that "Part II" of my notes (which specifically discuss differential geometry) are the toughest part of the notes.

Probably, I should mention a few examples of manifolds and explicitly study their coordinates in lecture 5.

3.1. Functions. I never discussed what it means for a function on a manifold (a) to exist, (b) to be smooth.

Really, this let us discuss curves too. Why? A curve is just a function γ:I→M where I is just a closed interval, and M is the manifold.

3.2. Diffeomorphisms. This word is thrown around a lot, but never defined rigorously (or at all!). So I should re-investigate this a bit.

Vertex algebras

Think about your favorite Lie algebra Lie(G). We have a mapping on it, namely, the adjoint representation:

ad:Lie(G) → End[Lie(G)]

where "End[Lie(G)]" are the endomorphisms of the Lie algebra Lie(G).

Normally this is of the form "ad(u)v∈Lie(G)" and is shorthand for "ad(u)=[u,-]".

The Jacobi identity looks like:

ad(u)ad(v)-ad(v)ad(u)=ad(ad(u)v).

This is the most important identity. Vertex operator algebras are an algebra with a similar property.

A vertex operator algebra consists of a vector space V equipped with a mapping usually denoted

Y:V→(End V)[[x,x^-1]].

In this form, it looks like left-multiplication operator...or that's the intuition anyways. So if "v∈V", we should think Y(v,x) belongs to "(End V)[[x,x^-1]]" and acts on the left.

Really through currying this should be thought of as "V⊗V→V[[x,x^-1]]", i.e., a sort of multiplication operator with a parameter "x". (This is related to the "state-operator correspondence" physicists speak of with conformal field theories.)

Just like a Lie algebra, the Vertex Operator algebra satisfies a Jacobi identity and it is the most important defining property for the VOA.

Lets stop and look at this structure again:

Y:V→(End V)[[x,x^-1]].

What's the codomain exactly? Well, it's a formal distribution (not a mere formal power series!).

So what does one look like? Consider δ(z-1) = Σ zⁿ where the summation ranges over n∈ℤ. This series representation is a formal distribution, and behaves in the obvious way. Lets prove this!

Desired Property: δ(z-1) vanishes almost everywhere.

Consider the geometric series f(z) = Σzⁿ where n is any non-negative integer (n=0,1,...).

Observe that δ(z-1) = f(z) + z^-1f(z^-1). Lets now substitute in the resulting geometric series:

δ(z-1) = [1/(1-z)] + z^-1[1/(1-z^-1)]

and after some simple arithmetic we see for z≠1 we have δ(z-1)=0.

Desired Property: for any Laurent polynomial f(z) we have δ(z-1)f(z)=δ(z-1)f(1).

This turns out to be true, thanks to the magic of infinite series; but due to html formatting, I omit the proof. The proof is left as an exercise to the reader (the basic sketch is consider δ(z-1)zⁿ, then prove linearity, and you're done).