A principal ${\displaystyle G}$ -bundle, where ${\displaystyle G}$  denotes any topological group, is a fiber bundle ${\displaystyle \pi :P\to X}$  together with a continuous right action ${\displaystyle P\times G\to P}$  such that ${\displaystyle G}$  preserves the fibers of ${\displaystyle P}$  (i.e. if ${\displaystyle y\in P_{x}}$  then ${\displaystyle yg\in P_{x}}$  for all ${\displaystyle g\in G}$ ) and acts freely and transitively (meaning each fiber is a G-torsor) on them in such a way that for each ${\displaystyle x\in X}$  and ${\displaystyle y\in P_{x}}$ , the map ${\displaystyle G\to P_{x}}$  sending ${\displaystyle g}$  to ${\displaystyle yg}$  is a homeomorphism. In particular each fiber of the bundle is homeomorphic to the group ${\displaystyle G}$  itself. Frequently, one requires the base space ${\displaystyle X}$  to be Hausdorff and possibly paracompact.

Since the group action preserves the fibers of ${\displaystyle \pi :P\to X}$  and acts transitively, it follows that the orbits of the ${\displaystyle G}$ -action are precisely these fibers and the orbit space ${\displaystyle P/G}$  is homeomorphic to the base space ${\displaystyle X}$ . Because the action is free and transitive, the fibers have the structure of G-torsors. A ${\displaystyle G}$ -torsor is a space that is homeomorphic to ${\displaystyle G}$  but lacks a group structure since there is no preferred choice of an identity element.

An equivalent definition of a principal ${\displaystyle G}$ -bundle is as a ${\displaystyle G}$ -bundle ${\displaystyle \pi :P\to X}$  with fiber ${\displaystyle G}$  where the structure group acts on the fiber by left multiplication. Since right multiplication by ${\displaystyle G}$  on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by ${\displaystyle G}$  on ${\displaystyle P}$ . The fibers of ${\displaystyle \pi }$  then become right ${\displaystyle G}$ -torsors for this action.

The definitions above are for arbitrary topological spaces. One can also define principal ${\displaystyle G}$ -bundles in the category of smooth manifolds. Here ${\displaystyle \pi :P\to X}$  is required to be a smooth map between smooth manifolds, ${\displaystyle G}$  is required to be a Lie group, and the corresponding action on ${\displaystyle P}$  should be smooth.

Trivial bundle and sections edit

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

Other examples edit

• 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.

Trivializations and cross sections edit

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

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

Characterization of smooth principal bundles edit

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 ${\displaystyle P/G}$ .

Reduction of the structure group edit

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:

• 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.

Associated vector bundles and frames edit

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.

Any topological group G admits a classifying space BG: the quotient by the action of G of some weakly contractible space, e.g., a topological space with vanishing homotopy groups. The classifying space has the property that any G principal bundle over a paracompact manifold B is isomorphic to a pullback of the principal bundle EG → BG.^[5] In fact, more is true, as the set of isomorphism classes of principal G bundles over the base B identifies with the set of homotopy classes of maps B → BG.

• Associated bundle
• Vector bundle
• G-structure
• Reduction of the structure group
• Gauge theory
• Connection (principal bundle)
• G-fibration

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• Jost, Jürgen (2005). Riemannian Geometry and Geometric Analysis ((4th ed.) ed.). New York: Springer. ISBN 3-540-25907-4.
• Husemoller, Dale (1994). Fibre Bundles (Third ed.). New York: Springer. ISBN 978-0-387-94087-8.
• Sharpe, R. W. (1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. New York: Springer. ISBN 0-387-94732-9.
• Steenrod, Norman (1951). The Topology of Fibre Bundles. Princeton: Princeton University Press. ISBN 0-691-00548-6.