DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 141–147

GEOMETRY OF COMPACT SYMMETRIC SPACES

MAKIKO SUMI TANAKA

Abstract. We determined g-signatures of compact symmetric spaces. Relating with this, we also determined the self-intersection of subspaces in compact symmetric spaces in several cases by using double fibrations.

1. Introduction

This is a joint work with Prof. Tadashi Nagano. We will discuss and determine the signature and g-signature of every compact oriented symmetric space M of dim M = 4n and for a orientation-preserving involution g of M. We will also discuss the self-intersection of a subspace of M in several cases. Our method is basically to apply the Atiyah-Singer theory [1], especially the g-signature theorem below to our geometric results ([3] and others). Informations on g-signature and on the self-intersection are reciprocal to some extent. The self-intersection is realized by a symmetric subspace of M and we uses double fibrations, G/K ∩ H → G/K and G/K ∩ H → G/H, to determine the self-intersections in some cases. Based on the found value of the self-intersection, we will state a geometric application concerning the geometric structures of exceptional symmetric spaces F II, EIII, EV I and EV III.

2. Preliminaries

Let M be a compact orientable 4k - dimensional Riemannian manifold. The signature τ(M) is the index of the quadratic form on H^{2k}(M : R) defined with the cup product and the orientation. It is reasonable to define τ(M) to be zero if dimM is not a multiple of 4. τ(M) and g − τ(M) are also defined as the indices of certain elliptic complexes. It is known that g −τ(M) is entirely determined by the action of g on the the real cohomology of M; hence it depends on the homotopy class of g only.

1991 Mathematics Subject Classification. 53C35.

Key words and phrases. Symmetric space, g-signature, self-intersection.

This paper is in final form and no version of it will be submitted for publication elsewhere.

Proposition 2.1 ([1]). The g -signature g−τ(M) equals the signature τ(M) if g is homotopic with 1_{M}, the identity map, or, more generally, if g acts on H^{2k}(M : R) as the identity.

Proposition 2.2. τ(M) is a certain Pontryagin number of M (the Hirzebruch index theorem). Hence (1) an m-fold covering space of M has the signature = mτ(M). (2) τ(M) = 0 if M is a group or M is a covering space of a nonorientable manifold. (3) τ(M × N) = τ(M) × τ(N); actually this product formula is valid in a much more general setting including the case where the dimension of M or N is not divisible by 4 and hence its signature equals by definition.

Proposition 2.3 ([1]). Let X be a compact oriented manifold of dimension 4k, and let g be an oriented preserving involution of X. Then one has

g − τ(X) = τ(SI(F (g, X); X)),

where F (g, X) is the fixed point set of g acting on X and SI(F (g, X); X) means the self-intersection of F (g, X) in X.

Proposition 2.4 ([2]). If the circle group U(1) acts on a compact oriented manifold X, then the signature τ(X) equals the sum of τ(F_{j}) for the connected components F_{j} of the fixed point set F (U(1), X) such that codimF_{j} ≡ 0 mod 4; F_{j} is oriented in a certain way by means of the action of U(1) as in [1]. The sum of τ(F_{k}) for F_{k} with codimF_{k} ≡ 2 mod 4 equals 0.

Proposition 2.5 ([4]). The signature of a quaternionic Kähler manifold X of dimension 4n is equal to b_{2n}, the Betti number of X at the half dimension 2n, up to sign.

Proposition 2.6. The signature of a symmetric Kähler manifold M is equal to P (i), the Poincaré polynomial P (t) with the imaginary unit i substituted.

We will work on the compact connected simple symmetric spaces unless otherwise mentioned. Such a space M is a homogeneous space G/K of a compact connected Lie group G which acts on M as isometries; the action is locally effective usually. M admits the point symmetry s_{o} for every point o of M, which is isometry and acts on the tangent space T_{o}M as −1 times the identity.

Proposition 2.7. s_{o} is homotopic to the identity 1_{M} if and only if (1) the Euler number χ(M) is positive, or equivalently, (2) the rank r(M) equals r(G) for M = G/K.

Corollary 2.8. Let s ∈ G have a finite order. Then one has the signature τ(M) = ∑ τ(F_{j}) for the connected components F_{j} of F (s, M) if F_{j} is oriented in an appropriate way.

Proof. s is then a member of some subgroup T ≅ U(1) of G. Since T acts on F (s, M) and F (T, M) is its subspace, one obtains the conclusion by using Proposition 1.4 twice; τ(M) equals the sum of the signatures of the components of F (T, M) as well as τ(F (s, M)). □

We call each component of F (s_{o}, M) a polar of o in M; the polars in M are concretely described in [3].

Corollary 2.9. (1) The signature τ(M) equals the sum of those of the polars of a point, if χ(M) is positive. (2) One has τ(M) = 0 otherwise. Proof. (1) is immediate from Corollary 1.8. In the case of (2), one has r(K) < r(G) and hence G contains a circle group T which fixes no point; and Proposition 1.4 gives τ(M) = 0. □

Remark 2.10. The polars in M are thus closely related to M in terms of topology. For another example, M is orientable if and only if every polars has an even dimension. Also the Lefschetz number Lef(s_{o}) equals the sum of their Euler numbers; hence, in case χ(M) is positive, the sum equals χ(M).

Proposition 2.11. s_{o} − τ(M) = τ(M) for the point symmetry s_{o}, if dimM ≡ 0 mod 4.

Proof. This equality obtains if s_{o} acts on H^{2k}(M : R) trivially, where dimM = 4k, by the generalized Lefschetz fixed-point theorem of Atiyah-Singer. s_{o} acts trivially on H^{2k}(M : R) which consists of the G-invariant (or, equivalently, harmonic) p-forms on M, provided p is even. □

3. Determination of the signature

We will determine the signature τ(M) for every compact symmetric space M, which we may assume is simple and simply connected with τ(M) ≥ 0.

Proposition 3.1. τ(M) = 0 if M is a group. Proposition 3.2. τ(AI(n)) = τ(AII(n)) = τ(DIII(n)) = τ(CI(n)) = τ(F I)) = τ(EI) = τ(EIV) = τ(EV) = τ(EV II) = 0.

Proof. χ(M) = 0 for M = AI(n), AII(n), EI and EIV. DIII(n) admits an involution whose fixed point set is DIII(2) × DIII(n − 2) and DIII(2) ≅ S^{2}. Also CI(n) admits an involution whose fixed point set is CI(1) × CI(n − 1) and CI(1) ≅ S^{2}. τ(S^{2}) = 0 implies τ(DIII(n)) = τ(CI(n)) = 0. There is a space EV^{%} (resp. EV II^{%}) which is doubly covered by EV (resp. EV II) and is not oriented. It implies τ(EV) = τ(EV II) = 0. □

Proposition 3.3. τ(G_{p}(C^{n})) = τ(G_{p}(H^{n})) = χ(G_{p}(R^{n})).

Proof. One may use the involution s = I_{1} ∈ U(1) ⊂ U(n), such that F (s, G_{p}(C^{n})) = G_{p}(C^{n−1}) ⨿ G_{p−1}(C^{n−1}). Hence τ(G_{p}(C^{n})) = I_{1} − τ(G_{p}(C^{n})) = τ(G_{p}(C^{n−2})) + τ(G_{p−2}(C^{n−2}) = χ(G_{p}(R^{n})). G_{p}(H^{n}) admits an involution whose fixed point set is G_{p}(C^{n}), so we conclude τ(G_{p}(H^{n})) = τ(G_{p}(C^{n})) □

Proposition 3.4. (1) τ(G_{r}(R^{n})) = 0 if n or r(n − r) is odd. (2) If r = 2p and n = 2m, then τ(G_{2p}(R^{2m})) = χ(G_{p}(R^{n})).

Proof. (1) is the case when G_{r}(R^{n}) is the covering space of a non-orientable manifold. For (2), G_{2p}(R^{2m}) has an involution whose fixed point set is G_{p}(C^{m}), so we have τ(G_{2p}(R^{2m})) = τ(G_{p}(C^{m})) = χ(G_{p}(R^{n})). □

Proposition 3.5. τ(GI) = τ(F II) = 1.

Proof. The polar of GI is {o} ⨿ S^{2} · S^{2}. τ({o}) = 1 and τ(S^{2} · S^{2}) = 0 implies τ(GI) = 1. The polar of F II is {o} ⨿ S^{8}. τ({o}) = 1 and τ(S^{8}) = 0 implies τ(F II) = 1. □

Proposition 3.6. τ(EII) = 4. Proof. EII has an involution whose fixed point set is G_{4}^{o}(R^{10}) ⨿ DIII(5). τ(G_{4}^{o}(R^{10})) = 4 and τ(DIII(5)) = 0 implies τ(EII) = 4. □

Proposition 3.7. τ(EIII) = 3. Proof. EIII has an involution whose fixed point set is G_{2}(C^{6}) ⨿ S^{2} × G_{1}(C^{6}). τ(G_{2}(C^{6})) = 3 and τ(S^{2} × G_{1}(C^{6})) = 0 implies τ(EIII) = 3. □

Proposition 3.8. τ(EV I) = 7. Proof. EV I has involutions whose fixed point sets are {o}⨿G_{4}^{o}(R^{12})⨿S^{2}·DIII(6) and EII ⨿ EIII respectively. So τ(EV I) = 1 ± 6 + 0 = ±4 ± 3 must be 7. □ Proposition 3.9. τ(EV III) = 7. Proof. EV I has an involution whose fixed point set is EV I ⨿ S^{2} · EV. So τ(EV I) = 0 ± 7 = 7. □ Proposition 3.10. τ(EIX) = 8. Proof. EIX has an involution whose fixed point set is G_{4}^{o}(R^{16}) ⨿ DIII(8)^{%}. So τ(EIX) = ±8 + 0 = 8. □

Corollary 3.11. One has

3τ(M) ≤ χ(M)

if dimM > 0 and this is sharp. In particular, the equality τ(M) = χ(M) implies χ(M) = 0 = τ(M).

4. The g-signature of symmetric spaces

In this section the proofs are omitted and refer to [5] for the proofs. Proposition 4.1. Let X be an R-space in a Kählerian symmetric space M, F (t, M) = X. Then one has (1) the normal bundle NX is isomorphic with T X, and hence NX has the Euler number χ(X); (2) X is connected; (3) SI(X; M) = χ(X); and finally (4) t − τ(M) = χ(X).

In most cases, one has t − τ(M) = τ(M) in (4) above. Precisely, this is valid for (X, M) = (G_{r}(R^{2r}), G_{r}(C^{2r})), (G_{r}(H^{n}), G_{r}(C^{n})), (SO(n), DIII(n)), (T · AII(n), DIII(2n)), (T · AI(n), CI(n)), (Sp(n), CI(n)), (G_{2}(H^{4})^{%}, EIII), (F II, EIII), (T · EIV, EV II), and (AII(4)/Z_{2}, EV II). The other cases are (U(r), G_{r}(C^{2r})), (S^{p} · S^{q}, G_{2}^{o}(R^{p+q+2})), (X, X × X^{−}) where X is Kählerian and the diagonal of the product of X and its ”complex conjugate” X^{−}. In every case, one has t − τ(M) = Lef(t).

Case of M = G_{p}(C^{n}).

(1) g = κ, the complex conjugation. F (g, M) = G_{p}(R^{n}). g − τ(M) = χ(G_{p}(R^{n})).

(2) g = s^{II} and p is even, g ≅ κ. F (g, M) = G_{p/2}(H^{n/2}). g − τ(M) = χ(G_{p}(R^{n})).

(3) g = s_{o} ∘ κ ∘ ad(J_{p}), n = 2p. F (g, M) = OIII(p). g − τ(M) = 0.

(4) g = κ ∘ ad(J_{p}), n = 2p. F (g, M) = CI(p). g − τ(M) = 0.

(5) g = ad(J_{p}), n = 2p. F (g, M) = U(p). g − τ(M) = 0. Case of M = G_{p}^{o}(R^{n}).

(1) g = I_{1} ≃ I_{h}, h = odd. F (g, M) = G_{p}^{o}(R^{n−1}). g − τ(M) = τ(G_{p}^{o}(R^{n−2})).

(2) g = ad(J_{p}), n = 2p. F (g, M) = UI(p). g − τ(M) = τ(M).

(3) g = s_{o} ∘ ad(J_{p}), n = 2p. F (g, M) = O(p). g − τ(M) = 0. Case of M = G_{p}(H^{n}).

(1) g = ad(J_{p}), n = 2p. F (g, M) = UII(p). g − τ(M) = 0.

(2) g = s_{o} ∘ ad(J_{p}), n = 2p, g ≅ ad(J_{p}). F (g, M) = Sp(p).

g − τ(M) = 0. Case of M = CI(n).

(1) g = ad(I_{n}); g preserves the orientation iff n ≡ 0 or 3 mod 4. F (g, M) = UI(n). g − τ(M) = 0.

(2) g = ad((1_{2} ⊗ J_{p})K_{n}), n = 2p. F (g, M) = Sp(p). g − τ(M) = 0. Case of M = DIII(n).

(1) g = ad(I_{n}); g preserves the orientation iff n ≡ 0 mod 4. F (g, M) = SO(n). g − τ(M) = 0.

(2) g = ad((1_{2} ⊗ J_{p})K_{n}), n = 2p. F (g, M) = UII(p). g − τ(M) = 0. Case of M = AI(n).

(1) g = I_{p}, p = odd. F (g, M) = T · (AI(p) × AI(n − p)). g − τ(M) = 0.

(2) g = s_{o} ∘ ad(I_{1}). F (g, M) is the disjoint union of G_{k}(R^{n}) for 0 < k < n, k = odd. g − τ(M) = 0.

(3) g = s_{o} ∘ J, n = 2p. F (g, M) = CI(p). g − τ(M) = 0. Case of M = AII(n).

(1) g = κ, p = odd. F (g, M) = DIII(n). g − τ(M) = 0.

(2) g = s_{o} ∘ (I_{1} ⊕ I_{1}). F (g, M) is the disjoint union of G_{k}(H^{n}) for 0 < k < n, k = odd. g − τ(M) = 0.

As for the exceptional type spaces, we have the following proposition. Proposition 4.2. One has s_{I} − τ(M) = τ(M) for every simply connected space M = E_{6}/K, where s_{I} is an outer involution of E_{6} induced on the spaces EI through EIV; we may assume that s_{I} is the point symmetry at a point of EI, since the outer involutions of E_{6} are homotopic with each other.

5. Self-intersections of some subspaces

In this section the proofs are omitted and refer to [5] for the proofs.

The following are obvious facts: (1) One has SI(M; M) = 0 if 2dimN < dimM.

(2) One has SI(N; M) = g − τ(M) if N is a component of F (t, M) such that 2dimN = dimM and the other components have dimension < dimN, where t and g are orientation preserving involutions of M which are homotopic. (3) If the homology class [N] = 0 in H_{∗}(M; R), then SI(N; M) = 0 in H_{∗}(M; R). So we have the next proposition.

Proposition 5.1. SI(G_{1}(H^{3}); F I) = 0, SI(S^{2} · CI(3); F I) = 0, SI(S^{8}; F II) = 1, SI(G_{1}(H^{3}); F II) = 1, SI(F I; EI) = 0, SI(F II; EIV) = 0.

Proposition 5.2. Let s and t be involutions of G for the symmetric space M = G/K and G/H respectively, where K and H are the connected components of F (s, G) and F (t, G). Assume that (1) s commutes with t, (2) G/H has an equal rank to the subspace K/K ∩H and (3) dimG/K is less than 2dimH/K ∩H. Then one has

SI(H/K ∩ H; M) = H(0)/H(0) ∩ K

and is connected, where H(0) is the centralizer of a maximal torus A ⊂ G/H in H, A ∋ o = H(o).

Corollary 5.3. SI(CI(p) × CI(q); CI(p + q)) = CI(1) × … × CI(1) × CI(q − p) if p ≤ q ( CI(1) ≅ S^{2} ),

SI(G_{4}^{o}(R^{9}); F I) = G_{4}^{o}(R^{7}),

SI(G_{4}^{o}(R^{10}); EII) = G_{4}^{o}(R^{6}),

SI(F I; EII) = SI(EII; EV I) = SI(EV I; EIX) = G_{4}^{o}(R^{8}),

SI(DIII(5); EIII) = G_{2}(C^{4}),

SI(CI(4)^{%}; EII) = 4.

Corollary 5.4. Let M be a compact, simply connected and simple Lie group and B be its subspace with the same root system as M. Then one has

SI(B; M) = [the maximal torus] = 0 .

By using the result of the self-intersections, we characterize the geometric structure of the exceptional type spaces: F II ⊂ EIII ⊂ EV I ⊂ EV III. Let M = M_{r} be one of these spaces with rank r = 2^{a}, a ∈ {0, 1, 2, 3} and dimM = 16r. Then we have SI(M_{2/r}, M_{r}) = 3.

References

[1] Atiyah, M., Singer, I., The index of elliptic operators, III, Ann. of Math. 87(1968), 546–604.

[2] Hattori, A., Taniguchi, H., Smooth S^{1} -action and bordism, J. Math. Soc. Japan 24(1972), 701–731.

[3] Nagano, T., The involutions of compact symmetric spaces, Tokyo J. of Math. 11(1988), 57–79.

[4] Nagano, T., Takeuchi, M., Cohomology of quaternionic Kähler manifolds, J. Fac. Sci. Univ. Tokyo 34(1987), 57–63.

[5] Nagano, T., Tanaka, M. S., The involutions of compact symmetric spaces, IV, to appear in Tokyo J. of Math.

Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo Noda, Chiba, 278-8510, JAPAN E-mail: makiko@ma.noda.sut.ac.jp