Differential Forms and the Hodge Star

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Supersymmetric Quantum Mechanics Week 3 (-4?): Differential forms and Hodge star Some of these problems are inspired by (or taken from) Chapters 12-14 of Lee's Introduction to Smooth Manifolds. 1. Recall the definition of the exterior algebra Λ∗(V ) of a vector space V and the wedge product. k Show that vectors v1; : : : ; vk are linearly independent if and only if v1 ^ · · · ^ vk 6= 0 2 Λ (V ). https://en.wikipedia.org/wiki/Exterior_algebra 2. Recall the definition of differential k-forms, Ωk(M) on a smooth manifold M as sections of the exterior bundle, ΛkT ∗M. Describe all elements of Ω∗(S1), Ω∗(S1 × S1), and Ω2(R4). https://en.wikipedia.org/wiki/Differential_form#Intrinsic_definitions 3. Recall the definition of the exterior derivative, d:Ωk(M) ! Ωk+1(M). Give an explicit description of the maps d:Ω2(R4) ! Ω3(R4) and d:Ω3(R4) ! Ω4(R4) and show that d◦d = 0. Show that d ◦ d = 0 in general. https://en.wikipedia.org/wiki/Exterior_derivative 4. Recall the definition of an oriented manifold. Show (or look up) how degree k differential forms can be integrated over a k-dimensional manifold. Write down (but don't evaluate) some integrals over S1 and S3 ⊂ R4 from your answers in 2 and 3. http://www.map.mpim-bonn. mpg.de/Orientation_of_manifolds#Orientation_of_topological_manifolds 5. On a Riemannian manifold M, show that the metric g determines a unique inner product on ΛkT ∗M that on Λ1T ∗M = T ∗M is the dual of g. See https://en.wikipedia.org/wiki/ Exterior_algebra#Inner_product for some hints. 6. On an n-dimensional Riemannian manifold M, show that for each k with 0 ≤ k ≤ n, there is a map ?:Ωk(M) ! Ωn−k(M) satisfying ! ^ ?η = h!; ηidvol where h−; −i is the inner product on ΛkT ∗M from 5, and dvol is the Riemannian volume form. This map is called the Hodge star operator. Give an explicit description of ?:Ω2(R4) ! Ω2(R4) and ?:Ω3(R4) ! Ω1(R4). https://en.wikipedia.org/wiki/Hodge_star_operator 7. For k-forms ! and η on a compact manifold M without boundary, show that Z Z (!; η) = ! ^ ?η = h!; ηidvol M M defines an inner product on Ωk(M) that extends the inner product on C1(M) = Ω0(M) from last week. 8. Using the inner product from the previous problem, show that the adjoint of d is given by d∗ = (−1)n(k+1)+1 ? d?; and deduce that d∗ ◦ d∗ = 0. https://en.wikipedia.org/wiki/Hodge_star_operator# Codifferential 9. Show that ∆ = dd∗ +d∗d:Ωk(M) ! Ωk(M) extends the definition of the Laplacian on 0-forms to all differential forms. 1.

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