In the discussion of topological

manifolds, one often comes across the useful concept of starting with two manifolds M

_{1}, M

_{2} and building a new manifold from them, using the

product topology: M

_{1} × M

_{2}. A fiber bundle is a natural and useful generalization of this concept.

Intuitively, the product topology places a copy of M_{1} at each point in M_{2}. Alternatively, it places a copy of M_{2} at each point of M_{1}. An example of this is R^{2} = R × R, where we take a line R as our base, and place another line at each point of the base, forming a plane. We could formulate another example with M_{1} = S^{1}, a circle, and M_{2} = (0,1), a line segment. The product topology here just gives us a piece of a cylinder, since this is what we get when we place a circle at each point of a line segment, or place a line segment at each point of a circle.

A fiber bundle is an object that is closely related to this idea. In any local neighborhood, a fiber bundle just looks like M_{1} × M_{2}. Globally, however, a fiber bundle is generally not a product manifold.

The prototype example for our discussion will be the möbius band, as it is one of the simplest examples of a non-trivial fiber bundle. We can create the möbius band by starting with the circle S^{1}, and (similarly to the case with the cylinder) at each point on the circle we attach a copy of the open interval (0,1), but in a nontrivial manner. Instead of just attaching a bunch of parallel intervals to the circle, our intervals perform a 180° twist as we go around. This gives the manifold a much more interesting geometry, as the boundary consists of only one curve, and the band is no longer orientable (there is no "inside" face or "outside" face).

Now, if we look at the object we've formed, we note that locally, it is indistinguishible from the cylinder. That is, the "twist" in the möbius band is not located at any particular point on the band; it is entirely a global property of the manifold. Motivated by this example, we seek to generalize the language of product spaces, to include objects like the möbius band which are only *locally* a product space. This generalization is what we will come to know as a fiber bundle.

**The Informal Description**

When we build up the language to describe a fiber bundle, we want to think intuitively that a fiber bundle is "locally a product space" in the same sense that a manifold is "locally euclidean". Thus, our language describing fiber bundles will mimic our language of manifolds closely.

The fiber bundle itself will be called the total space, E. It will be constructed from a base manifold M, and a fiber F. In our examples of the cylinder and the möbius band, we call S^{1} the base manifold, and the interval (0,1) is the fiber. However, since globally the cylinder and the möbius band differ, we're definitely going to need some additional data to distinguish them.

We will find that for a general point q in E, we can directly associate this with a point p in the base manifold, M. However, we cannot directly associate q with a point in the fiber, F. This is the first sign of asymmetry between M and F, and it can be seen readily in the case of the mobius band:

Say we want to parameterize the möbius band by a point θ on the circle, and a real number f in the interval (0,1). Concretely, say θ = 0 and f = ¾. Now, transport the point around the möbius band by increasing θ and keeping f fixed at ¾. When θ → 2π, we should return to the same point q in E, since it corresponds to the same^{+} θ and f, and hence the same q. However, because of the inescapable twist in the mobius band, the point we return to is associated with θ = 0, f = ¼. Our parameterization for F somehow "flips" when we move one turn around the möbius band.

What you should take away from this is that parameterization for F only works in a local sense; not globally. In this way, it is a great deal like coordinate parameterization on a manifold. In a neighborhood of a point, we can parameterize points in E by (p,f), but when we go to another neighborhood, we use a different parameterization (p,f ').

How to express this mathematically? First, we associate each point in E with a point in M, which we *can* globally do. This association can be accomplished by a projection map π: E → M, which projects points q in E to their associated point p in M. This map is generally not one-to-one, of course; we want it to map entire fibers F_{p} to points p, capturing the fact that we're attaching a copy of the fiber F to each point p. We can enforce this condition by the requirement that the preimage^{++} π^{ -1}(p) = F, for each p in M. Now, we still want to locally parameterize these fibers, but leave the definition open to include different parameterizations of F for different neighborhoods. This is the part that may be familiar from the standpoint of coordinate parameterization.

When we defined coordinates on manifolds, it was accomplished by an open covering {U_{i}}, and a homeomorphism Φ_{i} for each U_{i} associating it with an open set of R^{n}. We will do something very similar in the case of fiber bundles. We take an open covering of M, {U_{i}}, and a set of smooth homeomorphisms, {Φ_{i}}, associating an open set of E given by π^{ -1}(U_{i}) with a product space. Formally,

Φ_{i}: U_{i} × F → π^{ -1}(U_{i})

Since we have a map Φ_{i} between π^{ -1}(U_{i}) and U_{i} × F, we can locally express points in E using points in U_{i} × F. We first project the point q in E to a point p in M, using π(q) = p, then find an open neighborhood U_{j} of p, then there is a corresponding map Φ_{j} which associates q with a point in U_{j} × F, i.e. a pair of points (p,f).

Now, it is possible that we could have chosen a different U_{k} about p, and thus a different map Φ_{k} associating it with a different point (p,f ') in U_{k} × F. This is fine, but we need to understand the relationship between f and f '. In other words, to distinguish the fiber bundle properly, we have to know about all possible choices of fiber parameterization. In the case of the cylinder, there was only one fiber parameterization, because the space was globally a product space. In the case of the mobius band, there are two possible parameterizations, and we can make the transformation explicit by f = 1 - f '. Neither parameterization f nor f ' works globally; we can cover the circle with two overlapping segments, and choose one parameterization for one segment, and the opposite for the other segment.

Changes in parameterization of the fiber are known as transition functions. These are written formally as t_{ij} = Φ_{i} • Φ_{j}^{-1}: U_{i} ∩ U_{j} × F → U_{i} ∩ U_{j} × F, so they may be thought of as smoothly carrying a point from one product space to another, in the overlapping region U_{i} ∩ U_{j}. However, there are more enlightening ways to look at the transition functions. First of all, note that they carry the point p to itself: (p,f) → (p,f '). Thus, it may be more enlightening to think of this as a set of maps from F to itself for each point p in the overlap. Symbolically,

t_{ij}(p): F → F

Now, notice that t_{ij} satisfies group axioms:

t_{ij}(p) [ t_{jk}(p) ] = t_{ik}(p) is a transition function.

t_{ii}(p) = (identity map) is a transition function

t_{ji}(p) = t_{ij}^{-1}(p) is a transition function.

Thus, we can now think of {t_{ij}(p)}_{All p,i,j} as a group. Specifically, since we are interested in smooth transition functions, we think of {t_{ij}(p)} as a lie group. This group is denoted G, and is called the structure group of the fiber bundle. Of course, we can't forget that it is also a map from the fiber to itself. This can be thought of as a realization of the group, i.e. an action of the group G on the set F of points in the fiber. This group action is also required to be smooth, since that was an original requirement on the transition functions. To summarize, t_{ij} is a map

t_{ij}: U_{i} ∩ U_{j} → G

into the structure group G which acts smoothly on the fiber F. The transition functions characterize the fiber bundle. In the case of the cylinder, the structure group is just the trivial group of one element, the identity. In the mobius band, the structure group is the group of two elements, Z_{2}, given by {1,a} where a^{2} = 1. In other words, we only have two parameterizations, and thus only one transition function other than the identity, which is its own inverse ( 1-(1-f) = f ).

In these two cases, the structure group has a discrete, finite number of elements, and thus the dimensionality of the lie group is zero. Keep in mind that these are very simple cases of fiber bundles, and generally the lie group consists of a continuous spectrum of transition functions; in other words, we call this a lie group for a reason. Also keep in mind that we have a choice when determining our structure group, since we don't have to use all of the elements. For example, if the fiber bundle is trivial, like the cylinder, we can use any group G we want, but only use the identity element when defining transition functions. However, it makes the most sense to choose the smallest group that is convenient for our purposes.

**The Bloated Non-Elegant Attempt at a Formal Description**

So, we have finally laid out all the pieces we need to describe a fiber bundle. Let's give a preliminary formal definition, before eventually refining it more nicely.

A Differential Fiber Bundle (E, π, M, F, G) consists of the following:

- A differential manifold E called the total space
- A differential manifold M called the base manifold
- A differential manifold F called the fiber
- A surjective map π: E → M, called the projection, such that π
^{ -1}(p) = F_{p}, the fiber at p in M.
- An open covering U
_{i} of M with a diffeomorphism Φ_{i}: U_{i} × F → π^{ -1}(U_{i}) called the local trivialization, with π(Φ_{i}(p,f)) = p
- A lie group, G, known as the structure group, which acts on the fiber F.
Finally, there is the requirement that the transition functions, Φ_{i} • Φ_{j}^{-1} = t_{ij}(p), are smooth and live in G, the structure group.

Φ_{j}(f) = Φ_{i}(t_{ij}(p)f).

So, the base manifold and fiber tell you exactly what the bundle looks like locally. At the level of the manifold M, open neighborhoods just look like pieces of R

^{n}, and M's transition functions tell us precisely how they are sewn together. At the level of the bundle E, open neighborhoods are just pieces of R

^{n} × F, and there is an additional sewing operation. We need to glue the fiber F

_{p} over p from the patch U

_{i} to the same fiber F

_{p} from the patch U

_{j}. The structure group gives you the additional information required to tell you how to "glue" the fibers together. In this light, a fiber bundle is often seen as a natural generalization of the very concept of a manifold.

**Special Types of Fiber Bundles**

In the general case of fiber bundles, F can be any differential manifold and G can be any lie group that acts on F. By adding further requirements, we can define certain special bundles.

**The Trivial Bundle**
Almost unnecessary to include, except that it draws the important connection that shows that bundles are generalizations of product manifolds. Simply put, a trivial bundle is a product manifold. The base and fiber are interchangeable, and the structure group is just the trivial group of one element. The trivial bundle can be covered with one patch; the entire manifold M_{1}. The "local trivialization" Φ is really a global trivialization, since Φ covers all of M_{1}. The trivial bundle can clearly be formed using any two manifolds as base and fiber.

**Vector Bundles**
A vector bundle is defined by two things: first, the requirement that F be isomorphic to R^{k}, i.e. that F is a vector space. Second, that the structure group acts linearly on the vector space. Since the structure group acts on F linearly and F is a vector space, the transition functions have a k-dimensional representation on the fibers. In other words, the transition functions can be represented by k × k matrices.

**Special Case: The Tangent Bundle**
For example, take the base manifold to be M, and the fiber at p to be the tangent space T_{p}M, which is indeed a vector space. The projection operator π sends T_{p}M → p. For the open covering, we can use the same coordinate patches {U_{i}} that we used to define M.

This space is known as the Tangent Bundle of M, E = TM. We have a local trivialization in any given patch, simply given by the coordinate representation of the vectors V^{i} in T_{p}M. In other words, the coordinate charts not only give us a local parameterization for M, they also give us a local parameterization for TM, i.e. the vector components.

In the neighborhood U_{A}, we use coordinates {x^{i}}, and V_{p} ∈ TM = V_{p}^{i} ∂/∂x^{i}, and in the the neighborhood U_{B}, we use coordinates {y^{j}}, and V_{p} ∈ TM = W_{p}^{j} ∂/∂y^{j}.

At the level of the manifold, the transition functions are given by x(y) and y(x), but at the level of the bundle, we see that V_{p}^{i} = W_{p}^{j} ∂x^{i}/∂y^{j}|_{p}.

The transition functions, t_{AB}(p) = ∂x^{i}/∂y^{j}|_{p}. This map can be thought of as an n × n matrix mapping the components {W^{j}} to the components {V_{i}}. Since we can arbitrarily write down coordinate transformations on M, we can construct any set of matrices to produce the transition functions t_{AB}. Thus, the structure group of the tangent bundle of an n-dimensional manifold is GL_{n}R, the set of all invertible n × n real matrices. The transition functions of the manifold itself produce the maps t_{AB} which glue the fibers together.

Vector bundles in general are quite useful, due to their concrete nature. They are often viewed as generalizations of the tangent bundle. In this light, we can use similar formalisms for defining parallel transport and curvature in vector bundles, mainly because we can still use objects with indices, like Γ^{α}_{βδ}.

**Principal Bundles**
A principal bundle is a fiber bundle in which the fiber over any point p ∈ M is a copy of the bundle's structure group, F = G. Since G is a lie group, it is a manifold by definition. The group G can act on itself by left-multiplication. For example, imagine placing a circle at each point in a manifold, but smoothly rotating the circles as you move around the manifold. The circle can be thought of as a copy of U_{1} (Since U_{1} is a circle when viewed as a manifold), and the structure group is also U_{1}; it can be considered the group of rotations of a circle, but in this context of principal bundles we are thinking of U_{1} as a group which rotates itself.

**The Frame Bundle**
Take a manifold M, and let the fiber over p ∈ M be the space of all ordered bases {e_{i}} for T_{p}M. An ordered basis provides a frame at p. So, we are looking at all frames {e_{i}} at each p ∈ M. The reason we choose "ordered" bases is so that we don't distinguish between two sets of bases {e_{i}} and {f_{j}} where the {e_{i}} are just a permutation of the {f_{j}}. This is not quite a vector bundle, because a given element of E is a set of n *linearly independent* vectors. The independence condition prevents the fiber from being a vector space. For example, there is no "zero" element in the frame bundle. Now, note that given any initial basis for T_{p}M, you can get to any other by operating on this basis by a suitable element of GL_{n}R:

f_{j} = g_{ij} e_{i}

Since the {f_{j}} are a linear combination of the {e_{j}}, and they are linearly independent since g is nonsingular, the {f_{j}} do indeed form a basis for GL_{n}R. Moreover, this exhausts the space of frames. If you provide me with a frame {h_{j}}, I can always express each h_{j} as a sum of basis vectors in my basis. This is equivalent to writing h_{j} = g_{ij} e_{i}, where g_{ij} is an invertible matrix.

Thus, by starting with any fiducial or "point-of-reference" basis, we can get all other bases by acting with elements of GL_{n}R. Then, we can just label a given frame by the element of GL_{n}R that got us there from the fiducial frame. In this way, the fiber of all frames is nothing but the group GL_{n}R! In other words, the frame bundle is equivalent to a principal bundle.

The equivalence we've just shown seems useful. Is there any way of naturally going the other direction? That is, can we produce some kind of useful fiber bundle from a principal bundle? The answer is yes, from a principal bundle we can build associated k-dimensional vector bundles, provided that G has a k-dimensional representation. This is useful, because vector bundles are the most concretely-defined fiber bundles.

**Associated Vector Bundles**
The basic idea of constructing associated vector bundles is as follows: Rip out the copy of G at each p ∈ M. Replace by a vector space V of dimension k. Find a k-dimensional representation ρ_{k} of G. Then choose the transition functions to be ρ_{k}(t_{ij}) = k × k matrices acting on V.

More formally, let V be a k-dimensional vector space on which g acts via a k-dimensional representation, ρ. Then, given a principal bundle P, define the associated vector bundle P ×_{ρ} V by starting with P × V and imposing the equivalence relation

(u,v) ~ (u • g^{ -1}, ρ(g)v)

This equivalence relation does indeed replace each fiber G with a copy of V. Let's say u ∈ P is written locally as (x,h) where x ∈ M, h ∈ G. Then:

(x, h, v) ~ (x, h • h^{ -1}, ρ(h)v) = (x, e, ρ(h)v).

Thus, we can always rotate h into the identity thereby effectively collapsing the G-fiber to a point. At the same time, ρ(h)v ∈ V, so the V-fiber persists. Hence, we've replaced the G-fiber with a V-fiber.

It is easily seen above that the transition functions are just ρ(t_{ij}).

So, the large-scale picture you should have in mind is of a single principal bundle, underneath which we place a multitude of associated vector bundles. For every matrix representation of the structure group G, there exists a unique associated vector bundle. This should give you an idea of why they are called "principal bundles".

**What Actually ***is* a Fiber Bundle? (The more elegant description)

As mathematicians, we are inclined to rigorously define the tools that we use. Specifically, where do they live, and what distingishes them? For a fiber bundle, we have not yet explained this. Is it the total space? Is it the collection of spaces? What is the specific object we are calling the "bundle", and how does it specify all of the underlying structure? This is a delicate question, which is why it has been put off until we could get a more intuitive conceptual picture (also, jrn's inquiries made me realize that this was a gaping hole in the writeup).

Upon close inspection, you may notice that the fiber bundle is entirely specified by the projection map, π, subject to a rigorous series of requirements. All the other objects are defined through π. E and M are its domain and range, and it is required that they are both differential manifolds. Since it is required that E is a differential manifold, it is assumed that its differential structure is already fixed, but this structure is subject to all the requirements in the definition. F is given by π's preimage of a point in M. Local trivializations Φ_{i} are required to exist and be compatible, but they play a similar role to that of local coordinate charts on M. Since we required that E has a specific topology and differential structure, the local trivializations are just all possible maps which are compatible with this structure. Once all trivializations are given, this implicitly defines the set of all transition functions, and hence the structure group, G. Thus, all the pieces are truly given by just the projection map, and for this reason, it is the projection map itself which is often referred to as the "fiber bundle".

It will sometimes be useful to deem two different bundles π_{1} and π_{2} "equivalent". To do so, we need to be sure first that the two total spaces E_{1} and E_{2} are equivalent differential manifolds, i.e. there exists a diffeomorphism f: E_{1} → E_{2}. However, there must be an equivalence of the maps as well, so that the base manifolds are the same. In other words,

f(π_{1}(p)) = π_{2}(f(p))

Since a map specifies a bundle, this diffeomorphism equates the two bundles.

**Sections**

A section S of a fiber bundle is a map from the base manifold into the total space, picking out a point on each fiber F_{p} over any point p on the base. It's possible to think of a section as a fiber-valued function defined on the base. Since the section just picks out a point on the same fiber, we can project back down and get to the original point:

π [ S(p) ] = p

A section of a tangent bundle is more commonly known as a vector field.

**Parallel Transport**

Fiber bundles generally run into the same issue of ambiguity that we saw in the tangent bundle. There is no natural way to compare points in different fibers. As such, there is no god-given notion of parallel or horizontal transport. As another way to put it, it is not as yet meaningful to speak of "constant" sections. Naively setting f = constant will not work, as this will give us a different value f ' = t_{ij}(p) • f ≠ constant in a different fiber parameterization. In contrast, there *is* a well-defined notion of vertical transport, since the projection map π will tell us that we are still at the same point in the base. To fix some notion of "horizontalism", we need additional information, and this turns out to be the most general notion of a connection. Connections lead to curvature, which really moves us outside the scope of this writeup.

**Physics**

All of classical physics (the four fundamental forces of nature) can essentially be presented geometrically, using the language of fiber bundles. The electromagnetic field, for example, is often presented in the form of a rank-2 antisymmetric tensor, F^{μν}. This tensor, it turns out, is just the curvature tensor of a U_{1} principal bundle. The strong and weak nuclear forces also have field strength tensors, which are really just curvature tensors of the SU_{2} and SU_{3} principal bundles. Gravity manifests itself in the form of curvature of the tangent bundle, or (if you prefer) the curvature of the base manifold. Matter is described by sections of associated vector bundles. The data needed to prescribe an associated vector bundle is given by a representation. In other words, a particle's electric charge, color and flavor are really described by the associated vector bundle in which the particle is defined. A section of an associated vector bundle is better known by physicists as a wavefunction.

^{+}Try not to be confused by the fact that θ = 0 and 2π at the same time. This is a manifold issue, not a bundle issue. Don't let it distract you from the real parameterization problem, that of the fiber.
^{++}Even when a map is not invertible, there is a well-defined notion of a preimage; that is, the space of all points q in E which map to the point p in M.