What is the idea behind a principal bundle? It is not very hard to define and seems to contain some sort of notion of symmetry of the base space.
From a "sheaf-like" perspective, a principal bundle appears to adjoin "geometric information" to points (the fiber) and then consider the symmetries of this information according the structure group.
Could someone clarify this?
On
Actually this is a comment, a quite restricted view of how I understood ( or what I do not beyond) fwiw.
On a regular orientable manifold of an $ R^2$ surface in $R^3$ well defined with Frenet-Serret frame.. let there be smooth geodesics drawn in all orientations sharing a common tangent plane and normal at a point $P$.
The concurrent set of radial filaments can be called a tangent bundle of fibers, local variable curvature and torsion (obtained by rotation about normal in common tangent plane at $P$) get defined by vectorial definitions of Frenet-Serret moving trihedron with dot/cross products, there exist directions of maximum and minimum normal curvatures as well as maximum and minimum geodesic torsions. The bundle gives complete two parameter geometrical desription at $P$ as it varies location on the manifold including all rotations in tangent plane including resultant variations in curvature and torsion.
Euler normal curvature with geodesic torsion are definable at any filament direction of the bundle and are representable on a Mohr Circle of plot axes $(\kappa_n,\tau_g)$ which is generalizable to higher dimensions.
$$ \kappa_n= \kappa_1 \cos^2 \psi + \kappa_2 \sin^2 \psi,\, \tau_g=(k_1-\kappa_2) \sin\psi \cos\psi $$
A two fold symmetry is an essential feature of the Mohr Circle, so rotating vector between them makes $2 \psi$ rather than $\psi$ serving to represent all tensors e.g., generalized stress, strain, curvature, moment of inertia etc.
The idea is very much what @TedShifrin hinted at in his comment: A "bundle" in general is something that "smoothly" (or at least continously) attaches some object to every point of your manifold. Classical examples include the tangent bundle of your manifold, its dual, exterior powers, tensor products of it, frame and orientation bundles of such vector bundles (i.e. to every point of the base space you associate the set of basis/orientations of the vector space corresponding to this point). Every covering of your manifold is a bundle with discrete fibres.
Sometimes these objects come with symmetries: Basis can be transformed into one another by linear maps, orientations can be flipped, points in a fibre can be permuted via the monodromy action. Once you formalise that idea you have a "$G$-bundle": The frame bundle is a $GL(n)$-bundle, the orientation bundle is a $\mathbb{Z}/2$-bundle, any covering is a $\pi_1(M)$-bundle.
When you think of covering spaces there is a special one: The universal covering. Principal $G$-bundles are in some sense the right generalisation of universal covering spaces to the setting of $G$-bundles. Important examples include classifying spaces.