The tangent bundle TM, cotangent bundle T*M, and the bundle of k-forms Λᵏ(T*M) are all examples of vector bundles over M. What distinguishes a vector bundle from a general fiber bundle?
AVector bundles have fibers that are vector spaces and transition functions in GL(n)
BVector bundles are always trivial (isomorphic to a product)
CVector bundles must have one-dimensional fibers
DVector bundles require a Riemannian metric on the base
A vector bundle has vector space fibers and linear transition functions. The GL(n) structure group preserves the vector space structure of fibers, so operations like addition of sections and scalar multiplication are well-defined globally. General fiber bundles have arbitrary fibers (circles, Lie groups, homogeneous spaces) with arbitrary structure groups. Not all vector bundles are trivial (TS² is not), fibers can have any dimension, and no metric is needed.
Question 2 Multiple Choice
A global section of the tangent bundle TM is a vector field. A vector bundle E → M admits a nowhere-zero global section if and only if...
AE is trivial
BThe Euler class of E vanishes (when defined)
CE has a connection
DM is compact
A nowhere-zero section exists if and only if the Euler class e(E) ∈ Hⁿ(M) vanishes (for oriented rank-n bundles). The Euler class is the primary obstruction to finding a nowhere-zero section. For the tangent bundle TM, e(TM) = χ(M) (the Euler characteristic), which is why S² (χ=2) has no nowhere-vanishing vector field but the torus (χ=0) does. A bundle admitting a nowhere-zero section can split off a trivial line bundle: E ≅ E' ⊕ ε¹. But having a nowhere-zero section does not make E fully trivial.
Question 3 Short Answer
The Whitney sum E ⊕ F and tensor product E ⊗ F of vector bundles over M are again vector bundles. How are their fibers related to the fibers of E and F?
Think about your answer, then reveal below.
Model answer: The fiber of E ⊕ F at a point p is the direct sum Ep ⊕ Fp (as vector spaces), with dimension rank(E) + rank(F). The fiber of E ⊗ F at p is the tensor product Ep ⊗ Fp, with dimension rank(E) · rank(F). The transition functions of E ⊕ F are block diagonal (g_E ⊕ g_F), and those of E ⊗ F are the tensor product (g_E ⊗ g_F) of the transition functions. These operations make vector bundles over M into a semiring (the Grothendieck group completion gives K-theory).
These algebraic operations on vector bundles parallel operations on vector spaces. The dual bundle E* has fibers (Ep)*, the determinant bundle det(E) = Λʳᵃⁿᵏ⁽ᴱ⁾(E) has one-dimensional fibers, and the endomorphism bundle End(E) = E* ⊗ E has fibers GL(Ep). The rich algebraic structure of vector bundles is the foundation of K-theory, a generalized cohomology theory that captures bundle topology.