Vector Bundles

Core Idea

A vector bundle is a fiber bundle whose fibers are vector spaces and whose transition functions are linear maps (elements of GL(n)). The tangent bundle, cotangent bundle, and their tensor products are the fundamental vector bundles in differential geometry. Sections of vector bundles generalize vector fields and differential forms. Connections on vector bundles extend covariant differentiation beyond the tangent bundle, enabling the study of curvature for any geometric vector bundle.

Explainer

A vector bundle of rank k over a manifold M is a fiber bundle π : E → M where each fiber Ep = π⁻¹(p) is a k-dimensional real vector space, and the transition functions g_αβ : U_α ∩ U_β → GL(k, ℝ) act linearly on the fibers. The vector space structure of fibers means you can add sections and multiply them by smooth functions — the space of sections Γ(E) is a module over C∞(M), just like the space of vector fields.

The tangent bundle TM (fibers are tangent spaces TpM) and the cotangent bundle T*M (fibers are cotangent spaces T*pM) are the primary examples. Their tensor products give all tensor bundles: T^{r,s}M = TM⊗r ⊗ (T*M)⊗s, whose sections are (r,s)-tensor fields. The bundle of k-forms Λᵏ(T*M) is a sub-bundle of T^{0,k}M. Riemannian metrics are sections of the symmetric part of T*M ⊗ T*M. Connections, curvature tensors, and all the objects of Riemannian geometry are sections of various vector bundles.

A connection on a vector bundle E is a generalization of the covariant derivative: ∇ : 𝔛(M) × Γ(E) → Γ(E) satisfying C∞(M)-linearity in the first argument and the Leibniz rule ∇_X(f·s) = X(f)·s + f·∇_X s in the second. Connections on E need not come from any metric — they are additional structure. The curvature of a bundle connection is F(X,Y) = ∇_X ∇_Y - ∇_Y ∇_X - ∇_{[X,Y]}, now an endomorphism-valued 2-form (a section of Λ²(T*M) ⊗ End(E)). This generalizes the Riemann curvature tensor, which is the curvature of the Levi-Civita connection on TM.

Characteristic classes are topological invariants of vector bundles computed from the curvature of any connection. The Chern-Weil theory produces closed differential forms from invariant polynomials applied to the curvature form, and their de Rham cohomology classes are independent of the connection chosen. Chern classes (for complex bundles), Pontryagin classes (for real bundles), the Euler class (for oriented bundles), and Stiefel-Whitney classes (mod 2) measure the topological twisting of the bundle. The nontriviality of TS² is detected by its Euler class e(TS²) = 2 ∈ H²(S²). These invariants are the primary tools in the topological study of manifolds and are central to modern mathematical physics (gauge theory, string theory, topological quantum field theory).