Examples
The prototypical example of a smooth principal bundle is the frame bundle of a smooth manifold M, often denoted FM or GL(M). Here the fiber over a point x in M is the set of all frames (i.e. ordered bases) for the tangent space TxM. The general linear group GL(n,R) acts freely and transitively on these frames. These fibers can be glued together in a natural way so as to obtain a principal GL(n,R)-bundle over 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 O(n). The example also works for bundles other than the tangent bundle; if E is any vector bundle of rank k over M, then the bundle of frames of E is a principal GL(k,R)-bundle, sometimes denoted F(E).
A normal (regular) covering space p : C → X is a principal bundle where the structure group acts on the fibres of p via the monodromy action. In particular, the universal cover of X is a principal bundle over X with structure group (since the universal cover is simply connected and thus is trivial).
Let G be a Lie group and let H be a closed subgroup (not necessarily normal). Then G is a principal H-bundle over the (left) coset space G/H. Here the action of H on G is just right multiplication. The fibers are the left cosets of H (in this case there is a distinguished fiber, the one containing the identity, which is naturally isomorphic to H).
Consider the projection π: S1 → S1 given by z ↦ z2. This principal Z2-bundle is the associated bundle of the Möbius strip. Besides the trivial bundle, this is the only principal Z2-bundle over S1.
Projective spaces provide some more interesting examples of principal bundles. Recall that the n-sphere Sn is a two-fold covering space of real projective space RPn. The natural action of O(1) on Sn gives it the structure of a principal O(1)-bundle over RPn. Likewise, S2n+1 is a principal U(1)-bundle over complex projective space CPn and S4n+3 is a principal Sp(1)-bundle over quaternionic projective space HPn. We then have a series of principal bundles for each positive n:
Here S(V) denotes the unit sphere in V (equipped with the Euclidean metric). For all of these examples the n = 1 cases give the so-called Hopf bundles.
Read more about this topic: Principal Bundle
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