348 K. IMPLICIT FUNCTION THEOREM
Let Pk denote the space of degree k homogeneous polynomials on JRn+l and
let 1-lk denote the space of degree k homogeneous harmonic polynomials on JRn+^1.
For k E NU {O} one has the orthogonal decomposition
(K.47) Pk+2 = 1-lk+ 2 E9 r^2 Pk·
Since Po = 1-lo and P 1 = 1-£ 1 , applying induction to (K.47) yields the following.
PROPOSITION K.19 (Space of homogeneous (harmonic) polynomials). For all
j E NU {O} we have the decompositions
2 2. 2 2.
(K.48a) P2j = 1-l2j E9 r 1-l2j-2 E9 · · · E9 r J- 1-l2 E9 r JHo,
(K.48b) P2J+1=1-l2j+i E9r^2 1-l2j-1 E9 · · · E9r^2 J -2 1-l 3 E9r 2. JH1,
which are orthogonal with respect to the £^2 inner product on s n for the functions
restricted to sn. That is, for any f E 1-lk and h E 1-le, where k -=/=-£, we have
Letr f lsn hlsn dμ9sph = 0.
Jsn
Pk ~ Ulsn : f E Pk} and ilk ~ { flsn : f E 1-lk} ·
Note that pk c pk+2· By the proposition, EBkENU{O} ilk = ukENU{O} Pk, where
the LHS is an orthogonal direct sum. On the other hand, by the Stone- Weierstrass
theorem, u kENU{O} pk is dense in C^00 (Sn). It follows that EBkENU{O} ilk forms a
complete orthogonal system in £^2 (Sn). As a consequence, we have
PROPOSITION K.20 (Eigenvalues of Sn). Fork EN U {0}, the k-th eigenvalue
.k of the Laplacian acting on functions on s n is given by
(K.49) >..k = k (k + n - 1)
and the corresponding eigenspace is 1-lk.
4.2. Elements of Hodge theory on Riemannian manifolds.Let (Mn, g) be a closed Riemannian manifold. Let d : [2P ( M) ---+ [2P+l ( M)
and 6: f2P(M)---+ [2P-^1 (M) denote the exterior derivative and its formal adjoint,
respectively, acting on differential forms on M, where f2P(M) ~ C^00 (APT* M) is
the space of C^00 p-forms. R ecall that the Hodge star operator (for p = 0,... , n)
- : APT M ---7 A n-pT M
is defined to be the unique linear isomorphism such that ('-y, ry) dμ 9 = 'Y /\ *'r/ for any
ry, 'r/ E APT* M: The operator 6 acting on f2P(M) may be written as
6a = (-ltp+n+l d a.
We say that a differential form a is closed if da = O; likewise, a differential form
(3 is said to be co-closed if 6(3 = 0. Note that a differential form is closed if and
only if its Hodge star is co-closed; i.e., da = 0 if and only if 6 (*a) = 0. Likewise,
6(3 = 0 if and only if d ( *f3) = 0.
Let 6.d = - ( d6 + 6d) denote the associated Hodge- de Rham Laplacian acting
on differential forms.^4 With respect to the £^2 -inner product (ry, w) L 2 ~ JM (TJ, w) dμ
we have the following properties. The Hodge- de Rham Laplacian is self-adjoint:
(6. da, f3h2 = (a, 6.df3) L2 for a, (3 E f2P(M)(^4) This is the opposite of the usual sign convention.