MATH 623: HARMONIC FORMS HW
C. ROBLES
Assume: that Mis an oriented, n-dimensional Riemannian manifold.
Let Ωp(M) denote the (globally defined) p-forms on M.
1.) Prove that ∗∗ = (−1)p(n−p).
2.) Suppose that Mis Euclidean space Rn. Prove that ∆ = −Pn
i=1 ∂/∂xi2on functions.
3.) Prove that ∗∆=∆∗.
Assume: that Mis compact.
4.) Prove that the inner product on Ωp(M)
hα, βi=ZM
α∧ ∗β
is symmetric and nondegenerate. (Hint: first show that if v , w ∈ΛpT∗
xM, then v∧ ∗w=
w∧ ∗v.)
5.) Prove that δ2= 0.
6.) Suppose that α, β ∈Ωn(M) and RMα=RMβ. Assume that Mis connected and prove
that α−βis exact.
7.) Prove that Gis a bounded, self-adjoint linear operator.
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