Editing Principal bundle (section)

==Examples==

=== Trivial bundle and sections ===
Over an open ball <math>U \subset \mathbb{R}^n</math>, or <math>\mathbb{R}^n</math>, with induced coordinates <math>x_1,\ldots,x_n</math>, any principal <math>G</math>-bundle is isomorphic to a trivial bundle<blockquote><math>\pi:U\times G \to U</math></blockquote>and a smooth section <math>s \in \Gamma(\pi)</math> is equivalently given by a (smooth) function <math>\hat{s}: U \to G</math> since<blockquote><math>s(u) = (u,\hat{s}(u)) \in U\times G </math></blockquote>for some smooth function. For example, if <math>G=U(2)</math>, the Lie group of <math>2\times 2</math> [[Unitary matrix|unitary matrices]], then a section can be constructed by considering four real-valued functions<blockquote><math>\phi(x),\psi(x),\Delta(x),\theta(x) : U \to \mathbb{R}</math></blockquote>and applying them to the parameterization

<math display="block">\hat{s}(x) = e^{i\phi(x)}\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}.
</math>This same procedure valids by taking a parameterization of a collection of matrices defining a Lie group <math>G</math> and by considering the set of functions from a patch of the base space <math>U\subset X</math> to <math>\mathbb{R}</math> and inserting them into the parameterization.

=== Other examples ===
[[File:Z2 principal bundle over circle.png|thumb|300px|Non-trivial '''Z'''/2'''Z''' principal bundle over the circle. There is no well-defined 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 <math>M</math>, often denoted <math>FM</math> or <math>GL(M)</math>. Here the fiber over a point <math>x \in M</math> is the set of all frames (i.e. ordered bases) for the [[tangent space]] <math>T_xM</math>. The [[general linear group]] <math>GL(n,\mathbb{R})</math> acts freely and transitively on these frames. These fibers can be glued together in a natural way so as to obtain a principal <math>GL(n,\mathbb{R})</math>-bundle over <math>M</math>.
* 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 tensor|metric]]. The structure group is the [[orthogonal group]] <math>O(n)</math>. The example also works for bundles other than the tangent bundle; if <math>E</math> is any vector bundle of rank <math>k</math> over <math>M</math>, then the bundle of frames of <math>E</math> is a principal <math>GL(k,\mathbb{R})</math>-bundle, sometimes denoted <math>F(E)</math>.
* A normal (regular) [[covering space]] <math>p:C \to X</math> is a principal bundle where the structure group
: <math>G = \pi_1(X)/p_{*}(\pi_1(C))</math>
: acts on the fibres of <math>p</math> via the [[Covering space#Monodromy action|monodromy action]]. In particular, the [[universal cover]] of <math>X</math> is a principal bundle over <math>X</math> with structure group <math>\pi_1(X)</math> (since the universal cover is simply connected and thus <math>\pi_1(C)</math> is trivial).
* Let <math>G</math> be a Lie group and let <math>H</math> be a closed subgroup (not necessarily [[normal subgroup|normal]]). Then <math>G</math> is a principal <math>H</math>-bundle over the (left) [[coset space]] <math>G/H</math>. Here the action of <math>H</math> on <math>G</math> is just right multiplication. The fibers are the left cosets of <math>H</math> (in this case there is a distinguished fiber, the one containing the identity, which is naturally isomorphic to <math>H</math>).
* Consider the projection <math>\pi:S^1 \to S^1</math> given by <math>z \mapsto z^2</math>. This principal <math>\mathbb{Z}_2</math>-bundle is the [[associated bundle]] of the [[Möbius strip]].   Besides the trivial bundle, this is the only principal <math>\mathbb{Z}_2</math>-bundle over <math>S^1</math>.
* [[Projective space]]s provide some more interesting examples of principal bundles. Recall that the <math>n</math>-[[sphere]] <math>S^n</math> is a two-fold covering space of [[real projective space]] <math>\mathbb{R}\mathbb{P}^n</math>. The natural action of <math>O(1)</math> on <math>S^n</math> gives it the structure of a principal <math>O(1)</math>-bundle over <math>\mathbb{R}\mathbb{P}^n</math>. Likewise, <math>S^{2n+1}</math> is a principal <math>U(1)</math>-bundle over [[complex projective space]] <math>\mathbb{C}\mathbb{P}^n</math> and <math>S^{4n+3}</math> is a principal <math>Sp(1)</math>-bundle over [[quaternionic projective space]] <math>\mathbb{H}\mathbb{P}^n</math>. We then have a series of principal bundles for each positive <math>n</math>:
: <math>\mbox{O}(1) \to S(\mathbb{R}^{n+1}) \to \mathbb{RP}^n</math>
: <math>\mbox{U}(1) \to S(\mathbb{C}^{n+1}) \to \mathbb{CP}^n</math>
: <math>\mbox{Sp}(1) \to S(\mathbb{H}^{n+1}) \to \mathbb{HP}^n.</math>
: Here <math>S(V)</math> denotes the unit sphere in <math>V</math> (equipped with the Euclidean metric). For all of these examples the <math>n = 1</math> cases give the so-called [[Hopf bundle]]s.