Math 546 Topology and Geometry of Manifolds Spring 2000
Assignment #8
Due 5/19/2000
I. Required problems.
1. Let Vbe an n-dimensional vector space.
(a) Show that covectors ω1,...,ω
kon Vare linearly dependent if and only if
ω1∧···∧ωk=0.
(b) Suppose {ω1,...,ω
k}and {η1,...,η
k}are two collections of independent cov-
ectors on V. Show that the collections have the same span if and only if for
some nonzero constant c,
ω1∧···∧ ωk=cη
1∧···∧ηk.
(c) If ω∈ΛkV,ωis said to be decomposable if ωcan be written ω=σ1∧···∧σk,
where each σiis a covector. Is every 2-covector on Vdecomposable? Your
answer will depend on the dimension of V. Give proof or counterexample.
2. Prove Lemma 11.10.
3. Define a 2-form Ω on R3by
Ω=xdy∧dz +ydz∧dx +zdx∧dy.
(a) Compute Ω in spherical coordinates (ρ, ϕ, θ) (see Example 3.3).
(b) Compute dΩ in both Cartesian and spherical coordinates and verify that both
expressions represent the same 3-form.
(c) Compute the restriction Ω|S2=ι∗Ω, using coordinates (ϕ, θ), on the open
subset where these coordinates are defined.
(d) Show that Ω|S2is nowhere zero.
II. Optional problems.
4. Let Vbe a finite-dimensional vector space. We have two ways to think about the
tensor space TkV: concretely, as the space of k-multilinear functionals on V;and
abstractly, as the tensor product space V∗⊗···⊗V∗. However, we have defined
alternating and symmetric tensors only in terms of the concrete definition. This
problem outlines an abstract approach to alternating tensors.
Let Adenote the subspace of V∗⊗···⊗V∗spanned by all elements of the form
α⊗ξ⊗ξ⊗βfor covectors ξand arbitrary tensors α, β,andletAkVdenote
the quotient vector space V∗⊗···⊗ V∗/A. Define a wedge product on AkVby
ω∧η=π(eω⊗eη), where π:V∗⊗···⊗V∗−→ AkVis the projection, and eω,eηare
arbitrary tensors such that π(eω)=ω,π(eη)=η. Show that this wedge product is
