Principal bundle

Over an open ball ${\displaystyle U\subset \mathbb {R} ^{n}}$, or ${\displaystyle \mathbb {R} ^{n}}$, with induced coordinates ${\displaystyle x_{1},\ldots ,x_{n}}$, any principal ${\displaystyle G}$-bundle is isomorphic to a trivial bundle

${\displaystyle \pi :U\times G\to U}$

and a smooth section ${\displaystyle s\in \Gamma (\pi )}$ is equivalently given by a (smooth) function ${\displaystyle {\hat {s}}:U\to G}$ since

${\displaystyle s(u)=(u,{\hat {s}}(u))\in U\times G}$

for some smooth function. For example, if ${\displaystyle G=U(2)}$, the Lie group of ${\displaystyle 2\times 2}$ unitary matrices, then a section can be constructed by considering four real-valued functions

${\displaystyle \phi (x),\psi (x),\Delta (x),\theta (x):U\to \mathbb {R} }$

and applying them to the parameterization

$${\displaystyle U=e^{i\phi (x)/2}{\begin{bmatrix}e^{i\psi (x)}&0\\0&e^{-i\psi (x)}\end{bmatrix}}{\begin{bmatrix}\cos \theta (x)&\sin \theta (x)\\-\sin \theta (x)&\cos \theta (x)\\\end{bmatrix}}{\begin{bmatrix}e^{i\Delta (x)}&0\\0&e^{-i\Delta (x)}\end{bmatrix}}.}$$

This same procedure by taking a parameterization of a collection of matrices defining a Lie group and by considering the set of functions from a patch to ${\displaystyle \mathbb {R} }$ and inserting them into the parameterization.

Non-trivial Z/2Z principal bundle over the circle. There is no obvious way to identify which point corresponds to +1 or -1 in each fibre. This bundle is non-trivial as there is no globally defined section of the projection π.

• The prototypical example of a smooth principal bundle is the frame bundle of a smooth manifold ${\displaystyle M}$, often denoted ${\displaystyle FM}$ or ${\displaystyle GL(M)}$. Here the fiber over a point ${\displaystyle x\in M}$ is the set of all frames (i.e. ordered bases) for the tangent space ${\displaystyle T_{x}M}$. The general linear group ${\displaystyle GL(n,\mathbb {R} )}$ acts freely and transitively on these frames. These fibers can be glued together in a natural way so as to obtain a principal ${\displaystyle GL(n,\mathbb {R} )}$-bundle over ${\displaystyle M}$.
• Variations on the above example include the orthonormal frame bundle of a Riemannian manifold. Here the frames are required to be orthonormal with respect to the metric. The structure group is the orthogonal group ${\displaystyle O(n)}$. The example also works for bundles other than the tangent bundle; if ${\displaystyle E}$ is any vector bundle of rank ${\displaystyle k}$ over ${\displaystyle M}$, then the bundle of frames of ${\displaystyle E}$ is a principal ${\displaystyle GL(k,\mathbb {R} )}$-bundle, sometimes denoted ${\displaystyle F(E)}$.
• A normal (regular) covering space ${\displaystyle p:C\to X}$ is a principal bundle where the structure group

${\displaystyle G=\pi _{1}(X)/p_{*}(\pi _{1}(C))}$

acts on the fibres of ${\displaystyle p}$ via the monodromy action. In particular, the universal cover of ${\displaystyle X}$ is a principal bundle over ${\displaystyle X}$ with structure group ${\displaystyle \pi _{1}(X)}$ (since the universal cover is simply connected and thus ${\displaystyle \pi _{1}(C)}$ is trivial).

• Let ${\displaystyle G}$ be a Lie group and let ${\displaystyle H}$ be a closed subgroup (not necessarily normal). Then ${\displaystyle G}$ is a principal ${\displaystyle H}$-bundle over the (left) coset space ${\displaystyle G/H}$. Here the action of ${\displaystyle H}$ on ${\displaystyle G}$ is just right multiplication. The fibers are the left cosets of ${\displaystyle H}$ (in this case there is a distinguished fiber, the one containing the identity, which is naturally isomorphic to ${\displaystyle H}$).
• Consider the projection ${\displaystyle \pi :S^{1}\to S^{1}}$ given by ${\displaystyle z\mapsto z^{2}}$. This principal ${\displaystyle \mathbb {Z} _{2}}$-bundle is the associated bundle of the Möbius strip. Besides the trivial bundle, this is the only principal ${\displaystyle \mathbb {Z} _{2}}$-bundle over ${\displaystyle S^{1}}$.
• Projective spaces provide some more interesting examples of principal bundles. Recall that the ${\displaystyle n}$-sphere ${\displaystyle S^{n}}$ is a two-fold covering space of real projective space ${\displaystyle \mathbb {R} \mathbb {P} ^{n}}$. The natural action of ${\displaystyle O(1)}$ on ${\displaystyle S^{n}}$ gives it the structure of a principal ${\displaystyle O(1)}$-bundle over ${\displaystyle \mathbb {R} \mathbb {P} ^{n}}$. Likewise, ${\displaystyle S^{2n+1}}$ is a principal ${\displaystyle U(1)}$-bundle over complex projective space ${\displaystyle \mathbb {C} \mathbb {P} ^{n}}$ and ${\displaystyle S^{4n+3}}$ is a principal ${\displaystyle Sp(1)}$-bundle over quaternionic projective space ${\displaystyle \mathbb {H} \mathbb {P} ^{n}}$. We then have a series of principal bundles for each positive ${\displaystyle n}$:

${\displaystyle {\mbox{O}}(1)\to S(\mathbb {R} ^{n+1})\to \mathbb {RP} ^{n}}$

${\displaystyle {\mbox{U}}(1)\to S(\mathbb {C} ^{n+1})\to \mathbb {CP} ^{n}}$

${\displaystyle {\mbox{Sp}}(1)\to S(\mathbb {H} ^{n+1})\to \mathbb {HP} ^{n}.}$

Here ${\displaystyle S(V)}$ denotes the unit sphere in ${\displaystyle V}$ (equipped with the Euclidean metric). For all of these examples the ${\displaystyle n=1}$ cases give the so-called Hopf bundles.

One of the most important questions regarding any fiber bundle is whether or not it is trivial, i.e. isomorphic to a product bundle. For principal bundles there is a convenient characterization of triviality:

Proposition. A principal bundle is trivial if and only if it admits a global section.

The same is not true for other fiber bundles. For instance, vector bundles always have a zero section whether they are trivial or not and sphere bundles may admit many global sections without being trivial.

The same fact applies to local trivializations of principal bundles. Let π : P → X be a principal G-bundle. An open set U in X admits a local trivialization if and only if there exists a local section on U. Given a local trivialization

${\displaystyle \Phi$ :\pi ^{-1}(U)\to U\times G}

one can define an associated local section

${\displaystyle s:U\to \pi ^{-1}(U);s(x)=\Phi ^{-1}(x,e)\,}$

where e is the identity in G. Conversely, given a section s one defines a trivialization Φ by

${\displaystyle \Phi ^{-1}(x,g)=s(x)\cdot g.}$

The simple transitivity of the G action on the fibers of P guarantees that this map is a bijection, it is also a homeomorphism. The local trivializations defined by local sections are G-equivariant in the following sense. If we write

${\displaystyle \Phi$ :\pi ^{-1}(U)\to U\times G}

in the form

${\displaystyle \Phi (p)=(\pi (p),\varphi (p)),}$

then the map

${\displaystyle \varphi :P\to G}$

satisfies

${\displaystyle \varphi (p\cdot g)=\varphi (p)g.}$

Equivariant trivializations therefore preserve the G-torsor structure of the fibers. In terms of the associated local section s the map φ is given by

${\displaystyle \varphi (s(x)\cdot g)=g.}$

The local version of the cross section theorem then states that the equivariant local trivializations of a principal bundle are in one-to-one correspondence with local sections.

Given an equivariant local trivialization ({U_i}, {Φ_i}) of P, we have local sections s_i on each U_i. On overlaps these must be related by the action of the structure group G. In fact, the relationship is provided by the transition functions

${\displaystyle t_{ij}:U_{i}\cap U_{j}\to G\,.}$

By gluing the local trivializations together using these transition functions, one may reconstruct the original principal bundle. This is an example of the fiber bundle construction theorem. For any x ∈ U_i ∩ U_j we have

${\displaystyle s_{j}(x)=s_{i}(x)\cdot t_{ij}(x).}$

If ${\displaystyle \pi :P\to X}$ is a smooth principal ${\displaystyle G}$-bundle then ${\displaystyle G}$ acts freely and properly on ${\displaystyle P}$ so that the orbit space ${\displaystyle P/G}$ is diffeomorphic to the base space ${\displaystyle X}$. It turns out that these properties completely characterize smooth principal bundles. That is, if ${\displaystyle P}$ is a smooth manifold, ${\displaystyle G}$ a Lie group and ${\displaystyle \mu :P\times G\to P}$ a smooth, free, and proper right action then

• ${\displaystyle P/G}$ is a smooth manifold,
• the natural projection ${\displaystyle \pi :P\to P/G}$ is a smooth submersion, and
• ${\displaystyle P}$ is a smooth principal ${\displaystyle G}$-bundle over ${\displaystyle P/G}$.

Given a subgroup H of G one may consider the bundle ${\displaystyle P/H}$ whose fibers are homeomorphic to the coset space ${\displaystyle G/H}$. If the new bundle admits a global section, then one says that the section is a reduction of the structure group from ${\displaystyle G}$ to ${\displaystyle H}$ . The reason for this name is that the (fiberwise) inverse image of the values of this section form a subbundle of ${\displaystyle P}$ that is a principal ${\displaystyle H}$-bundle. If ${\displaystyle H}$ is the identity, then a section of ${\displaystyle P}$ itself is a reduction of the structure group to the identity. Reductions of the structure group do not in general exist.

Many topological questions about the structure of a manifold or the structure of bundles over it that are associated to a principal ${\displaystyle G}$-bundle may be rephrased as questions about the admissibility of the reduction of the structure group (from ${\displaystyle G}$ to ${\displaystyle H}$). For example:

The frame bundle ${\displaystyle {\mathcal {F}}(E)}$ of the Möbius strip ${\displaystyle E}$ is a non-trivial principal ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$-bundle over the circle.

• A ${\displaystyle 2n}$-dimensional real manifold admits an almost-complex structure if the frame bundle on the manifold, whose fibers are ${\displaystyle GL(2n,\mathbb {R} )}$, can be reduced to the group ${\displaystyle \mathrm {GL} (n,\mathbb {C} )\subseteq \mathrm {GL} (2n,\mathbb {R} )}$.
• An ${\displaystyle n}$-dimensional real manifold admits a ${\displaystyle k}$-plane field if the frame bundle can be reduced to the structure group ${\displaystyle \mathrm {GL} (k,\mathbb {R} )\subseteq \mathrm {GL} (n,\mathbb {R} )}$.
• A manifold is orientable if and only if its frame bundle can be reduced to the special orthogonal group, ${\displaystyle \mathrm {SO} (n)\subseteq \mathrm {GL} (n,\mathbb {R} )}$.
• A manifold has spin structure if and only if its frame bundle can be further reduced from ${\displaystyle \mathrm {SO} (n)}$ to ${\displaystyle \mathrm {Spin} (n)}$ the Spin group, which maps to ${\displaystyle \mathrm {SO} (n)}$ as a double cover.

Also note: an ${\displaystyle n}$-dimensional manifold admits ${\displaystyle n}$ vector fields that are linearly independent at each point if and only if its frame bundle admits a global section. In this case, the manifold is called parallelizable.

If ${\displaystyle P}$ is a principal ${\displaystyle G}$-bundle and ${\displaystyle V}$ is a linear representation of ${\displaystyle G}$, then one can construct a vector bundle ${\displaystyle E=P\times _{G}V}$ with fibre ${\displaystyle V}$, as the quotient of the product ${\displaystyle P}$×${\displaystyle V}$ by the diagonal action of ${\displaystyle G}$. This is a special case of the associated bundle construction, and ${\displaystyle E}$ is called an associated vector bundle to ${\displaystyle P}$. If the representation of ${\displaystyle G}$ on ${\displaystyle V}$ is faithful, so that ${\displaystyle G}$ is a subgroup of the general linear group GL(${\displaystyle V}$), then ${\displaystyle E}$ is a ${\displaystyle G}$-bundle and ${\displaystyle P}$ provides a reduction of structure group of the frame bundle of ${\displaystyle E}$ from ${\displaystyle GL(V)}$ to ${\displaystyle G}$. This is the sense in which principal bundles provide an abstract formulation of the theory of frame bundles.