The frame bundle F(M) of an n-manifold M has as its fiber over p the set of all ordered bases (frames) for TpM. What is its structure group?
AO(n), the orthogonal group
BGL(n, ℝ), the general linear group
CSL(n, ℝ), the special linear group
DSO(n), the special orthogonal group
The frame bundle F(M) is a principal GL(n, ℝ)-bundle. GL(n) acts on frames by changing basis: if (e₁,...,eₙ) is a frame and A ∈ GL(n), then (e₁,...,eₙ)·A = (Aⁱ₁eᵢ,...,Aⁱₙeᵢ) gives another frame. This action is free and transitive on each fiber (any two frames are related by a unique invertible matrix). Choosing a Riemannian metric reduces the structure group to O(n) — the orthonormal frame bundle is a principal O(n)-bundle. An orientation further reduces to SO(n).
Question 2 True / False
A connection on a principal G-bundle is a g-valued 1-form on the total space (where g is the Lie algebra of G). This is equivalent to specifying a horizontal distribution in the total space.
TTrue
FFalse
Answer: True
A connection 1-form ω on a principal bundle P assigns to each tangent vector of P an element of the Lie algebra g. The kernel of ω at each point defines a horizontal subspace complementary to the vertical subspace (the tangent to the fiber). The horizontal distribution tells you how to lift curves from the base to the total space — this is parallel transport. The curvature 2-form Ω = dω + ½[ω, ω] measures the failure of the horizontal distribution to be integrable (the Frobenius condition).
Question 3 Short Answer
How does the associated bundle construction relate principal bundles to vector bundles?
Think about your answer, then reveal below.
Model answer: Given a principal G-bundle P → M and a representation ρ : G → GL(V) of G on a vector space V, the associated bundle P ×_G V = (P × V)/G is a vector bundle over M with fiber V. The equivalence relation is (pg, v) ~ (p, ρ(g)v). Every vector bundle arises this way from its frame bundle. Conversely, given a vector bundle E, its frame bundle F(E) is a principal GL(n)-bundle, and E ≅ F(E) ×_{GL(n)} ℝⁿ via the standard representation. This correspondence makes principal bundles the universal framework for vector bundles.
The associated bundle construction is the bridge between principal bundles (where the group acts freely on fibers with no preferred point) and vector bundles (where fibers have linear structure). Different representations of the same group produce different vector bundles from the same principal bundle — this is how tensor bundles arise from the frame bundle via tensor representations of GL(n).