CHAPTER X

PRODUCTS OF SYMMETRIC SPACES

1. Guillemin rigidity and products of symmetric spaces

According to Proposition 10.3, with p = 0, the 1-form u on X satisfies the Guillemin condition.

2. Conformally flat symmetric spaces

3. Infinitesimal rigidity of products of symmetric spaces

Theorem 10.19. A product of Riemannian manifolds

where each factor Xj is either a projective space different from a sphere, or a flat torus, or a complex quadric of dimension 3, is infinitesimally rigid.

(ii) If k is even, then the section

of F_C does not satisfy the zero-energy condition.

LEMMA 10.28. Let k, l 1 be given integers and a1, a2 be given complex numbers. Suppose that k + l is odd and that the section

We now suppose that the geodesics and are equal to the geodesic defined above; then we have

for all Our hypothesis on h therefore implies that this last identity holds; thus we obtain the relation a1 + a2 + 8b = 0, which is also given by Lemma 10.27.

The previous discussion, together with the formulas (10.14), gives us the following result:

Since Ak is positive and fY (y) is non-zero, the vanishing of these integrals implies that a1 = a2 = 0.

Let k 0 be an even integer; the proof of the preceding lemma also shows that, if the section

PROPOSITION 10.34 and Lemma 2.6, together with the results stated above concerning the vanishing of certain G-modules of the form with , imply that the equality

holds. Thus we have proved that an even 1-form on the product which satisfies the zero-energy condition, is exact. As we have seen above, this result implies both Theorems 10.23 and 10.21.