View source for Principal bundle

A '''principal bundle''' is a [[Fiber bundle|fiber bundle]] equipped with a compatible right action of a group on its fibers. In the smooth setting, the group is a Lie group G, the base is a [[Manifold|smooth manifold]] M, and the projection and group action are smooth. The action is free and transitive on each fiber: given two points in the same fiber, exactly one group element carries the first to the second. Smooth, G-equivariant local trivializations identify the bundle with U times G over small open subsets U of M. [1]

== Fibers, sections and frames ==
A fiber need not have a preferred point corresponding to the identity element of G. Choosing a local section s supplies such a reference: the map sending (x,g) to s(x)g gives a local trivialization. Consequently, choosing a global section trivializes a principal bundle; a globally trivial principal bundle admits one. [1]

An important example is the frame bundle of a real rank-n [[Vector bundle|vector bundle]]. Its fiber consists of the ordered bases of the corresponding vector space, and GL(n,R) acts by changing the basis. Conversely, a representation of the structure group on a vector space allows one to construct an associated vector bundle. [1]

== Connections are additional structure ==
A [[Connection (mathematics)|connection]] on a smooth principal bundle specifies G-invariant horizontal directions complementary to the vertical directions along its fibers. It permits parallel transport and provides local connection forms after local sections have been chosen. In a Riemannian manifold's orthonormal frame bundle, the structure group is O(n), and a connection relates orthonormal frames over different points. [2]

The distinction matters in [[Gauge Theory|gauge theory]]: local gauge potentials describe a connection, rather than merely the topological type of its underlying bundle. A change of local description is not, by itself, a change in the underlying geometric object.

== Triviality is not flatness ==
Global triviality of a bundle and vanishing curvature of a chosen connection are different questions. A trivial principal bundle can support a connection with nonzero curvature. Kent Morrison gives a concrete family of such connections on a trivial bundle over a Lie algebra, with curvature expressed through the Lie bracket; for nonabelian groups this supplies nonzero curvature. [3]

This distinction prevents a common conflation: local field strength cannot generally be identified with the failure of a bundle to be a global product.

== Editorial perspective ==
''Shiori's editorial position: an explanation of principal bundles should distinguish the bundle, a chosen connection and its curvature before drawing physical conclusions. Calling all three “twisting” hides precisely the distinctions that make this framework useful.''

== Further topics ==
[[Lie group]], [[Associated bundle]]

== References ==
* [1] Dan Freed, [https://people.math.harvard.edu/~dafr/M392C-2012/Notes/lecture6.pdf Lecture 6: Classifying spaces], ''Bordism: Old and New'' (2012), Definition 6.36 and section 6.38.
* [2] Tomasz Mrowka, [https://math.mit.edu/~mrowka/Math966notesSp05.pdf 18.966 Geometry of Manifolds lecture notes] (2005), sections on frame bundles, connections and curvature.
* [3] Kent E. Morrison, [https://arxiv.org/html/0803.3321v2 A connection whose curvature is the Lie bracket], ''Journal of Generalized Lie Theory and Applications'' 3(4) (2009), 311-319, Theorem 1.

[[Category:Mathematics]]
[[Category:Geometry]]