Lesson 12 web

Lesson 12: Differential Geometry I

Introduction

Special relativity gave us a first glimpse of spacetime as a geometric arena: events, intervals, four-vectors, and tensors that transform according to the Lorentz group. To go further—whether toward general relativity, continuum mechanics on curved surfaces, or any theory written in coordinate-independent language—we need the systematic geometry that makes those objects precise. That geometry is differential geometry, and this lesson is its first installment.

We begin at the foundations. Set Theory supplies the raw language of collections and maps; Basic Topology then tells us what it means for a space to be connected, open, or locally Euclidean. With those notions in hand we can define Differentiable Manifolds and Submanifolds, the spaces on which calculus can be done without a preferred set of coordinates.

Once a manifold is in place we study the maps that live on it: Smooth Functions and Differentiable Mappings. Differentiation itself is recast geometrically through Tangent and Cotangent Spaces, which let us speak of Vectors and One-Forms in a Manifold without ever choosing a basis. Assembling those objects pointwise produces Vector Fields.

A brief return to Linear Algebra Revisited prepares the ground for the full apparatus of Tensor Algebra—the same index gymnastics we began in spacetime, now freed from any particular metric or coordinate chart. Curvilinear Coordinates show how the abstract constructions look in the charts we actually compute with, and Fields on Manifolds collect scalars, vectors, and tensors into the objects that will later carry physical meaning.

By the end of the lesson the informal four-vectors and tensors of special relativity will have become special cases of a general geometric language: manifolds, tangent spaces, and tensor fields that can be written on any smooth space, flat or curved. That language is the one we will use from here onward.

Set Theory

In this lesson we are going to talk about manifolds and the structures that reside in them. Before we can understand manifolds we need to gain some basic terminology. Set theory constitutes the language of modern mathematics, including the differential geometry we will need to study.

Describing Sets

We begin by saying that we can collect some related things if they follow three axioms. The first axiom is that if x is a given mathematical object under study, it is either part of our collection or it is not. This tells us that for any such collection there is some rule defining what constitutes a member of the collection. We will call this axiom S1. The second axiom is that each member of a collection must be distinguishable from all other members, thus we only note a given member once. We will call this axiom S2. The third axiom is that no collection can be a member of itself. This axiom is called S3. Any collection for which S1, S2, and S3 hold true is called a set.

The things within a set are called elements. If we have the set A, then we can say that a is an element of A by writing a∈A. If a is not an element of A, then we write a∉A. We can say that a set is determined by listing its elements. Even if the set contains more than one of the same elements it is listed only once. When listing elements if a trend continues without end we will use the ellipses, …, to denote such. In this way we can write an infinite set. For example, we can write the set of natural numbers, N=(1,2,3,…}.

Another method is to first understand that all the elements, a, of the set, A, share some property or properties in common. We can symbolize such a property of an element of A by writing, p(a). Then we can construct the set by saying that A consists of all elements a such that p(a) is true. We can symbolize this using what is called set-builder notation.

(12.1)

Where the symbol of the vertical line represents the phrase, “...such that...”.

We can have more than one property that defines the elements of a set. Say that we have both the property p(a) and the property q(a), then we can write the set

(12.2)

You can have as many properties as necessary.

The number of elements in a finite set, S, is called the cardinality of the set, denoted .

If two sets have the same cardinal number, then we can compare the elements of the two sets, one by one. We say there is a one-to-one correspondence between the sets. Such sets are called equivalent, and we write this A ~ B.

Subsets

If all elements of a set, A, are also elements of another set, B, then we call A a subset of B. We use ⊆ as the symbol for a subset, thus we write,

(12.3)

If two sets are subsets of each other, then they are the same set, and we call them equal, a form of equivalence. We use the symbol = for equal sets,

(12.4)

Proper Subsets

If A⊆B and A≠B, then we say that A is a proper subset of B, we write this,

(12.5)

A set having no elements is called the empty set, or equivalently the null set, we write this using the symbol Ø. The empty set is a subset of every set.

One thought is based on the idea that every set has at least one subset, the empty set. Often a set will have many subsets. If we make a set of all of the subsets of a given set, we call that the given set’s power set. For example, the power set P of the set A is denoted P(A).

(12.6)

Representing Sets

Venn Diagrams

We can portray these ideas graphically by drawing sets as circles or ellipses. So we can draw the sets A and B.

Figure 12.1 The diagram for two sets.

Then we can say that A⊂B, and draw this.

Figure 12.2 The diagram for a subset.

Such drawings are called Venn diagrams.

Set Operations

Now that we know how to specify sets, how to determine if two sets are equal, and whether they are subsets of one another, it is natural to ask if there other ways to combine sets? It turns out that there are.

The first way to combine two sets is just to list all of the elements common to both sets. This is called a union of two sets, and we use the symbol ∪.

(12.7)

We can also draw the Venn diagram of the union.

Figure 12.3 The union of two sets.

Similarly, we can make a new set out of two sets by considering elements that the two sets have in common. This is called the intersection of two sets, symbolized by ∩.

(12.8)

If there is no intersection between two sets, then they are said to be mutually exclusive.

(12.9)

Another word for mutually exclusive sets is disjoint sets.

We can also draw the Venn diagram of the intersection,

Figure 12.4 The intersection of two sets.

We can write a new set made of all elements in one set that are not in another, this is called the difference between sets.

(12.10)

We can also consider a difference where we consider every element of the two sets except for their intersection. This is called the symmetric difference.

(12.11)

One concept that is tricky is when we make a list of all possible elements that can be considered for a situation. That list forms a set called the universe of discourse, or the universal set, we can denote this by the symbol U. Every set you are considering is then a subset of the universe of discourse. Of course, in more advanced set theory it is true that there is no universal set, but we will keep this idea for now.

The difference between a set and the universe of discourse is called a complement.

(12.12)

We can also list the elements of two sets in an ordered way; the first element of the first set and the first of the second set make a pair, the first element of the first set and the second element of the second set make a second pair, etc. This is called a Cartesian product,

(12.13)

Figure 12.5 The Cartesian product of two sets.

this can be written

(12.14)

We will often call this a product set.

Basic Topology

What is topology? Topology concerns itself with the properties of sets that remain unchanged while the sets change in certain ways. We then need to ask three questions: “What kinds of sets do we consider in topology?” “What constitutes a topological change in a set?” “What properties remain unchanged in a topological change of a set?”

Topological Spaces

We begin by addressing the first of our three primary questions of topology, “What kinds of sets do we consider in topology?” The answer to this requires that a set must have within it the idea of nearness and continuity. We begin with the set S and we state that there will be subsets of S.

The set of points in a space is called, reasonably enough, a point set. We will call S a point set.

We can have collections of sets, but we don’t want to call them sets of sets. Instead we call them families of sets. If we have a family of ten sets, each labeled U, we can index them using a subscript, . We can define a set I={1,2,…,10} and call this our index set. We can then define our family over the index set I by writing . As an example, we say that the third set of our family, according to our index set, is .

We can index our collection of subsets for the set S, and thus write the family of subsets .

It turns out that there are Four properties of these subsets:

T1: As always in set theory Ø is one of the subsets in the family . We could call this the element of .

T2: The totality of S is also in the family of .

T3: Any arbitrary union of members of is a member of . In this way we see that the family is closed under the union of subsets.

T4: The intersection of a finite number of members of is also a member of . Here again we see that the family is closed under the finite intersection of subsets.

Any collection of subsets having these properties are called open sets. We can collect the family of open subsets into a kind of structure . Any family of open subsets having T is called a topology.

A point set S together with a topology T, (S,T), is called a topological space.

Why does physics need open subsets? We never have exact measurements. There is always an error associated with such measurement. Thus no model based on such measurements can be exact. The quantities will be exact within a certain error, and any value of a quantity within that error is acceptable. Thus every model will actually involve collections of open subsets based on the different possible values within allowed uncertainties.

Neighborhoods

Given a point within the point set S, p∈S, where we have the topological space (S,T) and U⊂S containing an open subset belonging to T. If p is also an element of U, then we say that U is a neighborhood of p.

Exercise 12.1: According to this definition, is U necessarily open?

Every point will have at least one neighborhood, the set S.

If a neighborhood happens to be open, then it is called an open-neighborhood.

If for some topological space (S,T), we have distinct points p and p' where then as neighborhoods such that . We call such a topological space Hausdorff. This is named for the German mathematician Felix Hausdorff (November 8, 1868 – January 26, 1942).

Metric Spaces

One example of a topological space is one where the idea of distance is formally established, what we call a metric space. The underlying topology is not only a family of open subsets, but also requires some distance function that exists between its points.

Any family having a finite number of sets within it, or an infinite number of sets where the index set is equal to the set of natural numbers, is called a countable family. Similarly, if a set contains a countable number of subsets then the set is called countable. An infinite set that has a higher-order infinite cardinal number is called uncountable.

Any point in a metric space will have a countable set of open neighborhoods with a special condition. That condition is that any set U that has the point p as an element will also have at least one as a subset. In other words, any subset of a metric space containing a point is a neighborhood of that point.

Exercise 12.2 Is this a property of topological spaces in general?

Topological spaces where this condition holds have a special title, they are said to be first-countable.

Topology Basis

How do we characterize a topological space? It seems that we will need to fix the underlying point set S and somehow determine what subsets it has that are to be considered as open. In fact, this raises a ticklish issue; the number of subsets is often infinite, and is often not even countable. How do we cope with this situation?

In a manner similar to vector spaces we can determine a subfamily of open subsets from which the whole can be recovered. Such a collection of open subsets B allow for any subset of T to be obtained as a union of elements of B.

How do we decide on what the elements of B are? We can apply this theorem,

Theorem 12.1: is a basis for T if and only if (from now on we will use the symbol iff for this) and such that .

I am not going to provide a proof for this theorem, but if you are interested consult a book on general topology.

Problem 12.1: Prove this theorem if you are interested.

No two distinct topologies will have the same basis. However, a fixed topology can have several bases. How can we determine if two bases correspond to the same topology? Here we have another theorem, stated without proof.

Theorem 12.2: B and B' are bases defining the same topology iff and then such that . This also applies to the reverse situation.

Exercise 12.3: Write the reverse situation for Theorem 12.2.

Problem 12.2: If you choose, write out a proof for Theorem 12.2.

If the family of subsets provided by Theorem 12.2 is countable then we say we have a countable basis. When a topology has a countable basis it is called second countable.

Exercise 12.4: Are second countable topologies always first-countable? Are first-countable topologies second-countable?

Exercise 12.5: If a topology is not first countable, can it be second countable?

This allows us to develop a hierarchy of topologies: We begin with topological spaces. If we say that a space is first countable then it is “promoted” to a first-countable space. It might then be a metric space, since all metric spaces are first countable. We can then test for second-countability, which not all metric spaces are.

If we have the topological space (S,T) with a family of open subsets whose union is (S), then we can call the family an open cover of S. If every open subset of S can be expressed as a union of subsets in the open cover, then the open cover forms a basis for the topology.

Networks

It is often the case that as we study the structure of topological spaces we will accept variations on a them. One example of such a variation is a small deviation from the notion of a basis that we call a network.

We can define such a network by saying that, in analogy to a basis, a network is a collection of subsets of the space such that any member of T can be gotten as a union of element of the network. The difference from a basis is that the subsets need not be open.

Mappings and Morphisms

We can state that there is some rule, φ, that connects two sets, A and B, so that an element of A is converted into a unique element of B. Such a rule is called a mapping, or function. We then can write φ:A→B. A mapping where no two elements of A are mapped to the same point in B, is said to be one-to-one or equivalently injective. Such a mapping could be called an injection. We say that the image of φ is φ[A]={φ(x)}x∈A}.

A mapping μ:A→B where, for every element in B there is a unique element in A, we say that μ is onto or a surjection.

A mapping that is both injective and surjective is called a one-to-one correspondence—or in less of a mouthful—a bijection. Such a mapping is said to be bijective.

It is often the case in mathematics that when we introduce some class of objects that we also introduce some structure-preserving mappings between objects of the same class. Such structure-preserving mappings are called morphisms. We have mappings between sets, we have linear mappings (also called linear transformations) on vector spaces, and so on.

Continuity

A mapping f between two topological spaces, f:S→T is said to be continuous at some point p∈S if where q=f(p) where .

Exercise 12.6: Show how this definition emulates the classical ε-δ definition of continuity.

It is important to always specify T when speaking of a continuous mapping.

Such a mapping is continuous if it is continuous at every point in its domain.

Is there a way of writing this where we may not have to prove it true for an uncountable set?

Yes, if the inverse images of open subsets of T are open in S.

In the 1960s Sir Christopher Zeeman (1925-2016) wrote a pair of seminal papers on the topology of spacetime. In this work every function on the light cone is continuous.

Exercise 12.7: Does this imply that Minkowski spacetime has a discrete domain space?

It is important to note that despite half a century passing, we still do not know the topology of spacetime. This is a good topic for research. Though I am not sure about the efficacy of some of the current research ideas where known physical quantities are assumed continuous and we then search for topologies that allow it.

Refinement

Assume that we have two topologies, and , defined on the same point set, S. If then there are two words we can apply to , the first is that it is weaker than and the second is that it is coarser. Similarly, we can say of that it is finer, stronger, or that it is the refinement.

If a mapping is continuous on a space having a topology T, then that mapping will be continuous on any refinement of T.

Topological Changes in Sets—Homeomorphisms

Say that we have the mapping φ between two topological spaces, φ:S→T. Let’s say that this mapping is a bijection.

Exercise 12.8: Explain what this means.

Let us further say that the bijection is continuous.

Exercise 12.9: Explain what this means.

If the inverse of the bijection is also continuous then the bijection is called a homeomorphism. In that case the two spaces are said to be homeomorphic.

This is a primary answer to the question, “What constitutes a topological change in a set?” Since morphisms are structure preserving, the third question we asked, “What properties remain unchanged in a topological change of a set?” becomes, “What properties remain unchanged under a homeomorphic mapping of a set?”

Exercise 12.10: If you already know what an equivalence relations is, show that a homeomorphism is an equivalence relation. If you do not, look up the meaning of an equivalence relation and then show it.

Closed Sets

The difference of a subset A with its topological space can be written S-A. We can classify a subset of a topological space as closed if its difference is open, that is (S-A)∈T.

Exercise 12.11: What does this mean?

It is important to note that the ideas of open and closed subsets are not mutually exclusive, that is a subset can be both open and closed, or neither.

Exercise 12.12: Is the empty set open, closed, both, or neither?

Exercise 12.13: In the topological space (S,T) is the point set S open, closed, both, or neither?

Let’s say we have some set A that is either open or closed. The smallest closed subset of A that contains A is called the closure of A and is denoted .

If we have a topological space (S,T), we can then consider some subset, U⊂S, where U need not be part of the topology, then there will be some closed sets that contain U. The intersection of all closed sets containing U, , is the closure of U, .

Equivalently, we can write

The closure of a closed set A is A. If A is the closure of itself, then A is closed.

The largest open subset of A is called its interior and is denoted .

If we define the family of all open sets in A as , then .

Given the point set S is the set of points where A is an open neighborhood.

The difference, is called the boundary of A and is denoted b(A).

We can also write .

If U is an open set of T, then and .

If U is a closed set, then and .

This last leads to a way of checking as an open set is always disjoint from its boundary, while a closed set always contains its boundary. A set that is both open and closed has an empty boundary.

From this we can think of the boundary of a set as its “shell.” A closed set includes its own shell.

Denseness

Say that we have two subsets, A and B, of a topological space (S,T). The subset A is said to be dense in B if .

The same subset will be everywhere dense if .

Exercise 12.14: The set of rational numbers is denoted Q. Is Q dense on ? Can this result be extended to the set of n-tuples of rational numbers in ?

A set is nowhere dense when the interior of its closure is empty.

Exercise 12.15: What does this mean?

Coverings

Given a topological space (S,T). A family of subsets of the point set S is called a cover if .

If all are open sets of T then the cover is called an open cover.

If we have an open cover, U, then we can also have a subcollection of U. If such a subcollection is also a cover then we call this a subcover.

A covering where each point has a neighborhood where the neighborhood intersects only a finite number of elements of the cover is called a finite cover.

A covering is a refinement of if such that .

Compactness

If every cover of a topological space has a finite subcover, then the topological space is called compact.

Exercise 12.16: What does this mean?

There are times in theoretical physics where a compact space makes things simpler. We can study situations in field theory where we make the assumption that the field will approach some asymptotic form that, hopefully, corresponds to the vacuum case at spatial infinity. Since all points at infinity are mapped to a point, we are effectively compactifying some non-compact space to a compact space. Since this compactifies the space based on a single point, we call it a one-point compactification.

The image of a compact space by a continuous function is compact.

Exercise 12.17: Does this work for inverse images?

A topological space is called locally compact if every one of its points has a neighborhood having a compact closure.

Exercise 12.18: What does this mean?

Exercise 12.19: Is every compact space locally compact?

Exercise 12.20 Is every locally compact space also compact?

A subset of a point set is called relatively compact if its closure is compact.

In this way a point set is locally compact if every point has a relatively compact neighborhood.

It is interesting that the idea of compactness places an imposed limit on the number of open sets. Though a space that is too fine-grained will violate its limitation.

Exercise 12.21: Why does compactness limit the number of open sets?

Connectedness

Given the open subsets of the point set, S, say A and B, where the intersection is disjoint, A∩B=Ø. If, further, we can write A∪B≠S, then we say that the topological space (S,T) is connected. If a space is not connected it is called disconnected.

This is equivalent to saying that no proper subset is both simultaneously open and closed.

The image of a connected space by a continuous mapping is connected.

Exercise 12.22: Does this work for inverse images?

For a point set S, if then such that f:[0,1]→S with f(0)=x and f(1)=y, then the topological space (S,T) is called arcwise connected. In most cases arcwise connectedness is equivalent to connectedness. A way of thinking about this is that we can make a continuous curve between any two points in a space where the entire curve is contained in the space defined as a loop.

Exercise 12.23: Does connectedness imply arcwise connectedness.

If this continuous map has the additional property that f(0)=f(1), then the map is called a loop. If, in a topological space, the loop can be continuously shrunk to a point, then the space is said to be simply connected.

Exercise 12.24: What does this mean?

Separation

If a topological space (S,T) has a subset that is everywhere dense, then that space is called separable. One way of thinking about this is there is a countable set of points in the point set S, and each open set in the topology contains at least one element of the countable set of points.

In a metric space this notion of separability is equivalent to saying that the metric space is second countable.

The more open sets we add to a topological space the easier it will be to separate its points. You can think of it like this, some point p is different than the point q as p∈U and q is not. Where points in the same neighborhood are practically indistinguishable. It can be a lot of work to make this precise for any given problem. It is likely that the separation properties of a space might only be determined by experiment.

To be clear, in the first instance we have two points that have their own neighborhoods that do not contain the other, but these neighborhoods need not be disjoint. A space meeting this case is called first-separable. If we require that the neighborhoods be disjoint, then the spaces are second-separable. This latter is equivalent to a space being Hausdorff.

Normal Spaces

If a topological space is both first-countable and Hausdorff then it is also called normal. In this way, every normal space is Hausdorff, but not every Hausdorff space is normal.

Exercise 12.25: Explain this.

Every metric space is normal, but there are normal spaces whose topology is not metrizable.

From this we state the following theorem without proof.

Theorem 12.3: This is also called Urysohn’s Metrization Theorem: A topological space that is second-countable is a metric space iff it is normal.

In other words, to show that a topological space is not metric, it suffices to show that it is not normal.

Paracompactness

A topological space is called paracompact if it is Hausdorff and all of its coverings have finite refinements.

Exercise 12.26: What does this mean?

In this way we can see that the existence of finite subcovers leads compactness. The extension to include finite refinement and Hausdorff character leads to paracompactness.

We can now extend our idea of the hierarchy of topological spaces. We begin with the topological space. Then we include first-countable spaces. Then we include Hausdorff spaces. Then we have normal spaces. Then we have paracompact spaces. And then we have metric spaces.

Every paracompact space is normal.

Exercise 12.27: Why is paracompactness important for integration?

We will see that paracompact spaces are essential to field theory, where linear connections to the space are represented by Christoffel symbols that we introduced in the Lesson 8.

Exercise 12.28: Does the spacetime interval on a Hausdorff space imply its paracompactness?

Topological Invariants

We can, somewhat dictatorially, state that if two spaces have different topological invariants then they cannot be homeomorphic. What is a topological invariant? Our dictatorial statement gives us a structural answer that is technically correct, but it doesn’t help us to understand it. A topological invariant is any quantity that is conserved under a homeomorphism.

There is no better way to understand it than that. It is possible to discover topological invariants by experimentation.

There are, of course, some examples of known topological invariants. The number of connected components of a space is one. Other examples are connectedness, compactness, or being Hausdorff. If we propose a topological invariant we need to prove it to be so.

We know that we do not know all of the topological invariants. Thus it is possible for two spaces to share all known topological invariants and still not be homeomorphic. We are left as we began, “If two spaces have different topological invariants then they cannot be homeomorphic.”

Topological Manifolds

We will now take our first crack at understanding manifolds. This is based on the topology we have been studying. Of course it is a natural question to ask, “Why not just use a topological space?”

One requirement of the product space of classical mechanics and the spacetime of relativity is that it looks sort of like . So, our question becomes, “Does a topological space look like ?”

The answer is, not in the least. We require that our spaces in physics have fixed dimensionality. The three dimensional space is three-dimensional everywhere.

If our space is a sphere, say we are representing the Earth. We cannot map its surface all in one piece onto a plane. We can slice it up and represent it piecewise on a plane.

This is where the idea of a topological manifold comes in.

To begin our hunt for what a manifold is, we begin by assuming that we have a topological space M.

Figure 12.6 Our prototype manifold.

We will also assume that we have an open subset of M, U.

Figure 12.7 An open subset in our prototype manifold.

We will then establish a mapping .

Figure 12.8 A mapping from the open subset in our prototype manifold to .

We will state our first two rules of manifolds,

C1: We will require that φ is a homeomorphism.

C2: φ[U]=O is open, .

Figure 12.8 The subset in mapped to by φ.

These structures can be combined, (U,φ), where we call such combinations n-charts or just charts. Another phrase used for a chart is a coordinate patch. The collection of all charts forms what we call an atlas.

We can then define an n-dimensional topological manifold as a second-countable Hausdorff space whose points have a neighborhood homeomorphic to an open subset in . The pair of the open subset and the homeomorphism forms a chart on the manifold.

Exercise 12.29: Explain what this means.

The Discrete Topology

Given a point set S we can declare every subset to be open. If we have the topological space (S,P(S)), where the topology is formed by the power set P(S). This is the definition of the discrete topology. We can define a set with a single element as a singleton. Thus every singleton {x} is a neighborhood of x. Thus the discrete topology is Hausdorff.

Exercise 12.30: Why?

Every subset in the discrete topology is both open and closed.

Exercise 12.31: Why?

Any mapping from the point set to any topological space is automatically continuous.

We have been referring to Zeeman’s attempt at the topology of Minkowski spacetime, this topology is discrete.

Differentiable Manifolds and Submanifolds

A mapping from a subset of to is a fancy way of saying that you now have n functions of n real variables. If all of the partial derivatives of these n functions exist and are continuous, we say that such a mapping is or smooth. So now we know what smoothness means. To apply this idea to what we have already written about manifolds, it tells us that a manifold is some space whose local structure is represented by n functions of n variables whose partial derivatives exist and are continuous on . The spacetime manifold of Lesson 11 is one such example of this.

It seems that we might be able to use these charts to somehow make sure that our space M has a local smoothness structure. Why? Such a chart sets up a correspondence between the point set U⊂M and the open point set . In other words a chart defines n real-valued functions on U. These functions have all relevant partial derivatives and those are . Such a set of functions is what we call the smoothness structure. The points of U can be labeled by the values of these n functions. When charts overlap we will further require that they share the same smoothness structure. We will require that we are able to establish an atlas on M.

Say that we have two charts on M, (U,φ), and (U',φ'). If U and U' intersect in M, then their intersection, U∩U', will induce two smoothness structures on M. One smoothness structure comes from φ and defines a bijection between the intersection U∩U' and the image of that intersection in , . Another smoothness structure comes from φ' and looks a lot like the previous one; it defines a bijection between U∩U' and .

Figure 12.8 Establishing the smoothness structure of two mappings.

We can compose these mappings, .

Figure 12.9 Composing our two mappings.

We can also take its inverse .

Figure 12.10 The inverse composition of our two mappings.

Together these compositions form a bijection between φ[U∩U'] and φ'[U∩U']. It is this bijection that we can use to compare the smoothness structures of our proposed manifold. We now state two more rules:

Co1: φ[U∩U'] and φ'[U∩U'] are open subsets of .

Co2: The mappings and are .

If these conditions are met for a pair of charts, those charts are said to be compatible. For charts to be compatible we only require that they share the same smoothness property. In this way you can isolate a single smoothness structure. It is important to realize that (U,φ), and (U',φ') must be compatible if U and U' do not intersect.

Let’s say we have a set, M, and we have a family of n-charts over an index set i on M, . If the following four conditions are met, then we call M an n-dimensional differentiable manifold.

M1: Any two charts of M are compatible. Another way of saying this is that if two charts induce a smoothness structure in the same region of M then those structures must agree.

M2: The charts cover M. This is another way of saying that all of M has an induced smoothness structure.

M3: Any n-chart compatible with the charts in the collection of charts C of M, is itself a member of C. This condition makes sure we are not overburdened with structure by limiting the number of charts we can put on M. By this condition we allow all compatible charts and remove all other structures.

M4: If we have distinct points in M, say p∈M and q∈M, then there will exist charts, and such that and where .

When we speak of manifolds, we write the label we have assigned and the charts are implied. There are different, but equivalent names for an n-dimensional differentiable manifold: -manifold (smooth manifold), Hausdorff manifold, and (this last is a mouthful) an n-dimensional manifold without boundary that is not necessarily paracompact or connected. In fact, the condition M4 prevents non-Hausdorff manifolds.

Exercise 12.32: Why?

Given any space M with a set of n-charts C, such that C satisfies M1, M2, and M4, then M with C is a manifold.

Exercise 12.33: Show this last sentence to be true. If true we can always reduce the structure to that of a manifold by including more charts.

For example, say we have M as the set of n-tuples of real numbers. We can consider M a point set, so that we write . Let U be any subset of that is open. Let be the identity mapping. (U,φ) is a chart (check C1 and C2 to make sure). We see that this satisfies M1, M2, and M4, and that makes this a manifold. M3 is a lot harder to implement, but fortunately we do not need it.

As another example assume that M is the set of (n+1)-tuples of real numbers that also satisfy the equation . Define a point set having . Define the mapping as acting on the point , transforming to . Then we define another point set having . Define the next mapping acting on the point , this transforms to . Define a point set having . We define the next mapping that acts on the point , becoming . If we continue in this way we will find (2 n+2) charts, and , where i=1,2,…,(n+1). These n charts are all compatible. This produces an n-dimensional manifold called the n-Sphere, .

Exercise 12.34: Show that is indeed a manifold by this definition.

Assume we have two manifolds, M and N of dimensions m and n respectively. We can define a new manifold, P, called the product of M and N where we write, P=M×N. The dimensions of P will be m+n. P is a point set made up of the pair (p,q) where p∈M and q∈N. How do we introduce the required charts for this to be a manifold? Since M is a manifold we have a chart where and . Similarly for N, we have a chart where and . Since P=M×N, then we have the open subset . Now we need to establish our mapping. We use the points we defined above. We have which maps to . We also have which maps to . This leads us to map to , thus establishing . Thus we have . This gives us a chart on P, thus P is a manifold.

For example. we can take the product of the manifolds . We can see this as extending the 1-sphere, , along the set of real 1-tuples, . The resulting manifold is called the cylinder manifold.

Exercise 12.35: Draw a diagram of and satisfy yourself that it is a cylinder.

As another example, we can take the product of the manifold . Here we see that we are extending the 1-sphere, , along another 1-sphere, . The resulting manifold is called the torus manifold.

Exercise 12.36: Draw a diagram of and satisfy yourself that it is a torus.

We have previously defined an open set. We can extend this definition to cases where we have a chart. A set O is open in M if for any point p∈O, there is a chart (U,φ) where p∈U⊂O.

We can also construct new manifolds by cutting holes in existing manifolds, but only certain hole-cutting is allowed. Assume we have the manifold M and a subset of it, S⊂M. Here we state that S is open in M. Let C be closed and C⊂M. We now define N=M-C, with the charts (U,φ) on M where U∩C=Ø.

Exercise 12.37: Is N a manifold?

Before we move on, let me say that for all of the fuss we have made about how to establish a manifold, they are completely boring objects. They are a blank canvas having a smoothness structure. They only become interesting when we put something on the canvas. Of course, linking back to our spacetime manifold this smoothness structure is what establishes the geometry of spacetime. If we make the assumption that spacetime is the world we live in, then the manifold contains the entirety of the dynamics of our world in its geometry.

Smooth Functions and Differentiable Mappings

Let’s say we have a manifold M, and we have some real-valued mapping, f, on M. If we have a chart on M, say (U,φ), then we can say that f:U→U, in other words f is a mapping from U to itself. We can state that is defined on .

Exercise 12.38: Why can we state this?

A less formal way of saying this is that is a real-valued mapping of n-variables.

Exercise 12.39: Let f be a function on M. Also let be for a collection of charts, C, satisfying M2 above. Show that f is a smooth function on M.

This is a standard technique for defining something on a manifold. You use a chart to describe that particular something in terms of . We will see why by the end of this lesson.

The collection of smooth mappings on M is something that we will denote by the gothic F, F.

Exercise 12.40: Let be a function of m variables. Let represent a set of smooth functions on M, where the index set goes from 1 to m. Show that is a smooth function on M.

As stated above, smooth mappings on a manifold, M, characterize the manifold structure of M.

Given the manifold M. let C and C' be two sets of charts on M satisfying M1-M4. If every smooth mapping on (M,C) is also a smooth mapping on (M,C'), then C=C'.

Exercise 12.41: Prove this last sentence to be true.

Exercise 12.42: Attempt to redefine a smooth mapping in terms of charts.

The morphisms I want to present below are called differentiable mappings.

Let M and N be manifolds. Define the mapping μ:M→N. Assume there is a smooth mapping, f, on N. Then the composition f◦μ is a mapping on M.

Morphisms all share the property that a composition of morphisms is itself a morphism. We establish a theorem, this time with a proof.

Theorem 12.4: Given the manifolds M, N, and O with smooth mappings and . Then the composition is also a smooth mapping.

Proof of Theorem 12.4: Let g be a smooth mapping on O. We will show that the composition is a smooth mapping on M. Since g is a smooth mapping on O and is a smooth mapping, then is a smooth mapping on N. Since is a smooth mapping on N and is a smooth mapping, then is a smooth mapping on M. QED.

Two objects that are identical with regard to the definition of a structure being considered are called isomorphic. We say that two such objects share an isomorphism. We have already seen an example of isomorphisms of sets are what we have called bijections. Isomorphisms of vector spaces are simply called isomorphisms. Isomorphisms of manifolds are called diffeomorphisms. Let’s say we have two manifolds, M and N. Say that we have μ:M→N. If μ is bijective and is a smooth mapping, then we call μ a diffeomorphism. In such a case M and N are said to be diffeomorphic.

The composition of diffeomorphisms is a diffeomorphism.

Exercise 12.43: Prove this last sentence to be true.

Exercise 12.44: Prove that if two manifolds are diffeomorphic, they have the same number of dimensions.

Theorem 12.5: Let p∈M. There is an open set, O, containing p with the following property: given any point q∈O there will exist a diffeomorphism μ:M→M with μ(p)=q.

Proof of Theorem 12.5: Let (U,φ) be a chart, and assume p∈U. Define . Now choose ε>0, such that and when d(x,z)<ε then x∈φ[U]).

Figure 12.11 Establishing a mapping from the manifold to .

The subset is our candidate for the open subset of M. Here V is the collection of all points with d(x,z)<ε.

Figure 12.12 Establishing an open subset in the manifold and in .

Define a point q∈O such that where .

Figure 12.13 Establishing a mapping from open subset in in to the open subset in our manifold.

We can further make ε'=d(z,y)<ε.

So, can we establish a diffeomorphism that slides from one point to another nearby point?

We introduce a mapping l:V→V where and where we set r=d(x,z). Here f(r) has the properties:

r1: f(r) is .

r2: There exists such that f(r)=1 when .

r3: There exists such that f(r)=0 when .

r4: .

We can see a prototypical example of this,

Figure 12.14 A prototype cutoff that is identically 1 near the origin, and identically 0 past and a smooth translation between them as described by r1 through r4.

Exercise 12.45: Prove r1, r2, r3, and r4.

Since ε'<ε the function f(r) exists.

The mapping is smooth by conditions r1 and r2. We can see by r2 that l(z)=y. By all four conditions we can see that l is the identity near the edge of V. We can also see that exists and is smooth.

Exercise 12.46: Show that the assertions of the previous paragraph are true. If necessary get hold of a book on Advanced Calculus and follow through a proof of the inverse function theorem and adapt it to this version of the theorem.

We can then define μ:M→M. If we say that s∈M then we have two cases, s∈O or s∉O. If we accept the first case we write . If we accept the second case then we write μ(s)=s. It seems that μ is a smooth bijection, and that μ(p)=q. QED

Exercise 12.47: Prove that μ is a diffeomorphism.

Theorem 12.5 seems to allow us to use charts to pull some specific property of into a property of the manifold M.

What we have done here is develop the ability to make connections from our manifold to archetypal sets that maintain the geometry of the manifold in the form of diffeomorphisms. This sets the stage for our ability to place things on our blank canvas.

Bump Functions

The mapping f(r) that we constructed above is similar to another kind of function.

Assume we have an open set U. If we state that U has a compact closed cover K, then it is possible to find a smooth function that is 1 in K but falls off rapidly to 0 in U outside of K. This is called a bump function. Here is an example.

Figure 12.15 A prototype bump function.

Tangent Vectors and One-Forms in a Manifold

While in all generality we will be discussing geometrical objects throughout a manifold, it is convenient to begin by discussing objects at a point. To that end, let’s say we have a tangent vector located at a point, p, in denoted as . By establishing a local origin in we can treat it is if it were a local copy of , and this allows us to establish n-tuples Each element of the vector n-tuple is a component of the vector in . In this way the vector can be represented by its components with respect to a set of coordinate axes (a chart) in , . Another way of saying this is that a tangent vector in is equivalent to a list of n real numbers. This is a completely natural representation of a vector in with a local origin. It is not necessarily so natural in an arbitrary manifold—where there need be no “natural” axes.

Assume that we are in . The collection of all smooth mappings on will be denoted as . Thus an element of will be a real-valued mapping of n-variables, . Given a vector v having the set of components and also having a smooth mapping on , f, then we can define

(12.15)

This is called a directional derivative of f in the direction represented by .

We can see from the elementary properties of derivatives that our directional derivative v(f) will satisfy three conditions:

DD1: The Sum Rule: v(f+g)=v(f)+v(g)

DD2: The Power Rule: v(f g)=f(x)v(g)+g(x)v(f)

DD3: The Constant Function Rule: If f is a constant function, then v(f)=0.

We can see how this works with manifolds. Say we have our manifold, M. On this manifold we have a collection of smooth mappings, F(M). By the results from Exercise 12.39, DD1, and DD2 we can see that the pointwise sum and product of elements of F(M) are also elements of F(M). This gives F(M) the structure of a ring.

Exercise 12.48: Show this to be true.

Choosing a point, p, of M, then a tangent vector at p is a mapping , and this satisfies DD1, DD2, and DD3, where x is replaced by p. The collection of all tangent vectors in the manifold M located at p are denoted F(p). Our goal now is to show that F(p) is a vector space. To do that we begin by stating that v and w are in F(p). If we further state that we have some real number, say m, and an element of F, say f, then we can write (v + m w)(f)=v(f)+m w(f), and with this we are have accomplished our goal of establishing a vector space. We have shown how to add elements of the space and how to multiply elements by a real number.

Theorem 12.6: Equation (12.15) defines a bijection between the n-tuples and mappings from to . This bijection satisfies DD1, DD2, and DD3.

Proof of Theorem 12.6: There are two things that we need to show to prove this theorem. The first is that equivalent directional derivatives have equivalent components. The second is that mappings from to satisfy DD1, DD2, and DD3.

To prove the first part let v(f)=w(f) for all f∈F. We set . Then by (12.15) we have . Since we can do this for all we have proved the first part.

To prove the second part we state that there is a mapping, and that w(f) satisfies DD1, DD2, and DD3. We can define a set of n values . This allows us to refine what we are trying to do. We now need to show that for this set of values (12.1) holds for all f. To accomplish this let , we we then rewrite f,

(12.16)

where the set . To proceed we need to introduce a Lemma,

Lemma 12.7: If f is in then it can be written as (12.16) where p is some fixed point of for some .

Proof of Lemma 12.7: We can write,

(12.17)

We can then write .

QED.

Continuing from this point, we apply DD1 and DD2 to get

(12.18)

Since p is a fixed point we can use DD3 to state that μ[f(p)]=0.

(12.19)

By using both DD1 and DD3 , so

(12.20)

By (12.16) mwe can write , so

(12.21)

We can see that , so

(12.22)

This proves the second part. QED

Tangent vectors can also be represented in terms of charts. Given a chart on M, (U,φ), and where we label a smooth mapping f where the composition is a smooth mapping on the image φ[U]. If our point p is in U, then we can write p=φ(p). We now introduce a set of n functions of F such that in an open subset of having our point p we get . In this way, if v is a tangent vector in our manifold at a point, then the set of numbers are called the components of the tangent vector with respect to the chart (U,φ).

Exercise 12.49: What is the purpose of the h functions?

Theorem 12.8: For a tangent vector, the choice of components is independent of the choice of h functions. In fact, given a set of components there exists a unique tangent vector in a manifold at a given point with those specific components.

Proof of Theorem 12.8: Say that f is an element of the collection of smooth functions, F. If we also allow the functions , s, and t are also in F, such that s(p)=0. We can write f,

(12.23)

Exercise 12.50: Construct (12.23). Choose the set of functions so that on some open set containing p. Choose some s that is almost always positive, except that it vanishes at p. Choose t so that it satisfies (12.23).

Using an argument similar to that when we proved Theorem 12.6, and the fact that , we get

(12.24)

This proves our first statement, that the components are independent of the h functions.

The second statement of the theorem is proven by first noting that if we have a set of numbers , then (12.24) defines a tangent vector in M at the point p with the set of numbers as components. If M is an n-dimensional manifold, and p is a point in M, then the composition F◦(p) is an n-dimensional vector space.

QED

The components of a vector depend on our choice of chart. We derive the well-known formula for this dependence. Let (U,φ) and (U',φ') be two charts, both containing the point p of M. Then U U' is handled in two ways, once by φ and once by φ'. That is to say we have a smooth bijection φ◦φ' from φ[U U'] to φ'[U U'], both subsets of . This gives us a set of n mappings of n variables. We can write this set of mappings, . Now let f be a smooth mapping on M. We can write and . If v is a tangent vector in M at p and and are it components with respect to our two charts (U,φ) and (U',φ') respectively, then we can use (12.24) to conclude that

(12.25)

Since this must hold for all smooth mappings,

(12.26)

Tensor Algebra

Assume we have a set of finite-dimensional vector spaces over the field of reals, , , ... . Note that for this discussion we are assuming our indices are summed over all n-values of the spatial dimensions, even though we are using Latin indices. Assume we have a tangent vector, v, where we denote which vector space the tangent vector belongs to by specifying the index. If we say that v belongs to then we write . We can say that the tangent vector w belongs to by writing . We can only add tangent vectors if they have the same superscript.

Let the mapping be linear, that is—given some number m and two vectors, —then . Say we also have a linear mapping labeled . We then define a new linear mapping, . This definition of linear mappings forms a new vector space, we call it the dual of , denoted . Instead of writing we write , alternately we can write . We can see that a formula such as,

(12.27)

is of the correct form for introducing more mappings.

Choose a finite and ordered list of vector spaces such that a vector space appears only once in the list and that no vector space appears with its dual, so you cannot have , but you can write .

Establish a mapping .

If the mapping is linear in each variable on its own

(12.28)

the mapping, G, is called a multilinear mapping.

The set of all multilinear mappings on a product set of vectors spaces, say , will be denoted . We can call such spaces tensor spaces. An element of such will be denoted . We will ignore some notation we just introduced. Instead of writing , we will instead write . In a similar way, we will denote the sum of elements along with the product of an element with some number m, .

(12.29)

Thus our product set of vector spaces itself is a vector space. The elements of such tensor spaces are called tensors in the same way that vectors are elements of vector spaces.

As previously noted the superscript indices are called contravariant indices, the subscripts are called covariant indices. The total number of indices is called the rank of the tensor. A number is a symbol having no indices and is called a scalar, or equivalently a rank zero tensor. A symbol having one index is called either a tangent vector, or a one-form, or equivalently a tensor of rank one.

Say that we have two tensors that share no indices, . What happens when we multiply them together? We write the expression and this defines a multilinear map of the form . Thus we form the tensor space , the tensor element has the form . This operation is called the tensor product and you might recall it from previous lessons. We have taken a rank two tensor and a rank five tensor and made a rank seven tensor. For an n-dimensional tensor, the rank two tensor has elements, the rank 5 tensor has elements, and the tensor product has elements. This is why we use Mathematica, in spacetime this translates to 16, 1024, and 16,384 elements respectively. That is a lot of book-keeping. Of course we can often invoke symmetries to reduce the number of elements, sometimes drastically.

It turns out that multilinear mappings having the same vector space structure are isomorphic. So tensor spaces like and are isomorphic. An element of one tensor space is an element of any of them. This means that we can add tensors having the same index structure, say .

Think about using a swap here.

We can also substitute a different letter as an index so long as we maintain its position throughout our expression. For example if we substitute p for m in the expression we end up with .

Curvilinear Coordinates

We have already seen that a set of tangent vectors, , located at a point, p, on an n-dimensional manifold form an n-dimensional vector space. We can construct tensors over such a vector space. Such constructed tensors are called tensors on the manifold M at the point p.

Say that we have a chart (U,φ) where p∈(U,φ). Let’s further say that we have a one-form located at p. We can then write where is the tangent vector with components (1,0,…,0), called a basis, we also write where is the tangent vector with components (0,1,…,0), and so on.

Say that we have a second chart (U',φ') where p∈(U',φ'). If we let y=φ(p) and y'=φ'(p) (be careful, this is not the derivative). Then we can also write , similar to what we wrote in Lesson 11 in the section Special Relativity, Events, and Spacetime. If we think about it long enough we will find,

(12.30)

We can also write and with respect to our two charts. We then apply (12.26) and get,

(12.31)

but this only holds when

(12.32)

is true. Equation (12.32) is sometimes called the Transformation Law for a Fixed One Form.

Let’s say we have some quantity populating a field, Q. We now introduce some notation. The partial derivative of this quantity with respect to our charts is,

(12.33)

Say we have a point at some location in spacetime , what is Q as we pass to a neighboring point located at ? This small variation in Q will be,

(12.34)

If we make a small variation in we get the four values , these form the components of a tangent vector. If we change coordinates, we get,

(12.35)

This looks similar to (12.33). We can rewrite it in our new notation for (12.34)

(12.36)

So we can write the Transformation Law for Tangent Vectors

(12.37)

We can write out the chain rule this way

(12.38)

Make sure this is true

We can rewrite the transformation law for one forms

(12.39)

The transformation law for tensors can be written,

(12.40)

As an interesting way of categorizing tensors broadly if then we say the tensors are symmetrical.

If, instead, , then we say that the tensors are antisymmetrical.

If we have scalar field, s and we want to take its derivative,

(12.41)

thus, as we have already noted, the gradient of a scalar field is a one-form field.

Fields on Manifolds

A field, in the context of geometry and physics, is the assignment of a quantity or structure to each point in a manifold. You are, no doubt, familiar with scalar and vector fields. All tensors in a manifold will have the same index structure. So, if we have a tensor field on M, say , then for each point, p∈M, we will have the tensor . Given a specific basis in M the components of such a tensor field will be mappings of the n coordinates specified by the basis.

The operations we have studied so far, tensor addition, the tensor product, index substitution, and contraction can all be performed in a point-wise manner throughout a field. In this way such operations form their own tensor fields on the manifold.

Let be a tangent vector field on M. By definition this means that for every point p in M there exists a tangent vector . While it might be temping to say this is all we have to consider, think about the definition of a tangent vector from Lesson 4. Each chart establishes a set of smooth mappings, and our tangent vector assigns a real number to each such smooth mapping. This itself results in a mapping, f, at every point in M. We can label this function by the symbol , this function is sometimes called the Lie Derivative of the function f in the direction of . While we will not go into the properties of Lie derivatives here, we will list some of the properties that seem to be quite natural, and are merely extensions of the properties of a directional derivative.

VF1: The Sum Rule:

VF2: The Product Rule:

VF3: The Constant Function Rule: If f is a constant function, then .

We can then define a vector field as a mapping, from smooth mappings on M to functions on M that satisfy VF1, VF2, and VF3.

Let be a one form field on M. If is a tangent vector field on M, then is a mapping on M. In this way the one-form field defines a mapping of a tangent vector field on M to smooth mappings of M. If we have two tangent vector fields on M, say and , and h is a smooth mapping on M, then

(12.42)

Thus such a mapping is linear.

Now suppose that we have O as a mapping from tangent vector fields on M to functions on M. Say also that O is linear as defined in (12.42). In this case O assigns to each point in M a mapping from tangent vectors to numbers. In other words O defines a one-form field on M. We can now say that there is a bijection between one-form fields on M and linear mappings from tangent vector fields on M to smooth mappings on M. Thus we could define a one-form field as a linear mapping (by 12.42) from tangent vector fields on M to smooth mappings on M.

Now suppose we have a tensor field on M, say . Say we have the one-form and tangent vector fields on M, say , , and . Then is a smooth mapping on M. We then say that defines a mapping from one-form and tangent vector fields on M to smooth mappings on M. The mapping T is multilinear, if f is some smooth mapping on M we write,

(12.43)

The converse applies. A mapping from fields on M to smooth mappings on M that is multilinear (as in (12.43), defines a tensor field on M. Thus we can define a tensor field as a multilinear mapping form fields on M to smooth mappings on M.

The point, no pun intended, of all of this formalism is to say that we did not need to begin with tensors at points, we could have just defined tensor fields on the manifold.

What then constitutes a smooth field? A tangent vector field is smooth if for every smooth mapping f on it the Lie derivative of the function is smooth. A one-form field is smooth if, for every smooth tangent vector field , then is a smooth mapping. A tensor field is smooth if, for any smooth fields , is a smooth function. From now on we will stop adding the word smooth, and assume it unless otherwise stated. All fields will be thought of as smooth. If you want to try to do relativity on broken fields, go for it.

Doing this Stuff in Mathematica

We want to find out what Mathematica knows about topological spaces. We do Crtl= for entity discovery.

Doesn’t tell us much. Let’s try to get a description.

A topological space, also called an abstract topological space, is a set X together with a collection of open subsets T that satisfies the four conditions:

1. The empty set Ø is in T.

2. X is in T.

3. The intersection of a finite number of sets in T is also in T.

4. The union of an arbitrary number of sets in T is also in T.

Alternatively, T may be defined to be the closed sets rather than the open sets, in which case conditions 3 and 4 become:

3. The intersection of an arbitrary number of sets in T is also in T.

As you can see, Mathematica knows what a topological space is. This is an example of an Entity in Mathematica.

What happens when we list the kinds of topological spaces Mathematica knows about?

We can copy and paste one of these types to see more about it.

If we hadn’t discussed Hausdorff spaces, or if you don’t recall what it means we can see if Mathematica knows.

A topological space fulfilling the

-axiom: i.e., any two points have disjoint neighborhoods. In the terminology of Alexandroff and Hopf, a

-space is called a Hausdorff space. A

-space is sometimes said to “have Hausdorff topology” or “be Hausdorff.” An Etale space provides an example of a space that is not