Parallel Connections over Symmetric Spaces

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding Riemannian (or unitary) vector bundles with parallel curvature then reduces to finding representations of the structure group of the canonical principal bundle. This paper concerns connections on Riemannian vector bundles over simply connected symmetric spaces. Given a hypothesis about the Riemannian curvature of a symmetric space, we show that every Riemannian vector bundle with parallel curvature over that space is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding Riemannian vector bundles with parallel curvature then reduces to finding representations of the structure group of the canonical principal bundle. Our results apply also to unitary vector bundles, since a C bundle can be viewed as an R bundle together with an additional structure. The hypothesis we put on a symmetric space is quite mild, and is satisfied by all simply connected irreducible symmetric spaces, and by simply connected reducible symmetric spaces that do not contain an R factor. If a symmetric space does contain such a factor, then the theorem does not hold, and we give explicit counterexamples. Recall that a Riemannian connection on a vector bundle E → M is said to be Yang-Mills if its curvature tensor R is harmonic; i.e., if dAR E = 0, where dA is the covariant divergence operator. Yang-Mills connections have generated substantial interest, and much of our current knowledge of the topology of smooth 4-manifolds comes from their study [DK]. In the special case of the Levi-Civita connection on the tangent bundle of a Riemannian manifold, there is a stronger concept which has been of central importance in Riemannian geometry. A Riemannian manifold M , with Riemann curvature tensor R , is said to be locally symmetric if all covariant derivatives of R vanish. This is equivalent to R (X,Y )Z being a parallel vector field along every path γ for any X, Y , Z parallel along γ. This concept generalizes naturally to arbitrary Riemannian vector bundles E →M : Definition. A connection on E is said to be parallel if ∇R = 0; equivalently, if for any smooth path γ of M , parallel vector fields X, Y along γ, and parallel section U of E along γ, the section R(X,Y )U is parallel along γ. 1991 Mathematics Subject Classification. 53C05, 53C07, 53C20, 53C35. Typeset by AMS-TEX 1 2 LUIS GUIJARRO, LORENZO SADUN, AND GERARD WALSCHAP This condition involves, of course, the Riemannian metric on the base manifold M . (By a Riemannian vector bundle we mean a vector bundle with an inner product on each fiber. A connection on such a bundle is required to respect the inner product, meaning that the inner product is parallel. In this paper all vector bundles are assumed to be real and Riemannian, except where stated otherwise.) Recall that a symmetric space is a Riemannian manifoldM for which the geodesic symmetry at any point is a global isometry of M . Every locally symmetric space has a globally symmetric universal cover. Conversely, every symmetric space is locally symmetric. If M is a symmetric space, then M can be written as G0/H0, where G0 and H0 are groups, with H0 = Hol(M), the holonomy group of M . For example, S = SO(n+ 1)/SO(n) and CP = [U(n+ 1)/U(1)]/U(n). By lifting to the universal cover of G0 we can writeM = G/H, where G is simply connected and H is a cover of H0. If M is simply connected, then H is connected. For example, S = Spin(n+1)/Spin(n) and CP = SU(n+1)/U(n). (In the CP example, H0 and H are both isomorphic to U(n) as groups, but H is still the n+1-fold cover of H0.) Let Mp denote the tangent space of M at p. The space Λ(Mp) is naturally identified with so(Mp), the Lie algebra of skew-adjoint endomorphisms of Mp, by requiring that (1.1) u ∧ v (w) = 〈u,w〉v − 〈v, w〉u, u, v, w ∈Mp. The holonomy of M acts on Λ in a natural way via h(u ∧ v) = hu ∧ hv, for h ∈ Hol(M). Under the above identification, the action of Hol is by conjugation on so(Mp). This is also true for ΛE and the holonomy group in any vector bundle E with a Riemannian connection. Furthermore, the Riemannian curvature R at p is an equivariant map from so(Mp) to itself. It is easy to see that in general [kerR , ImR ] ⊂ kerR , and therefore the same is valid for the linear subspace spanned by [kerR , ImR ]. For our purposes, we will consistently assume that R satisfies a stronger condition: Condition A: span[kerR , ImR ] = kerR . This condition is well understood; in fact, in §4 we will prove: Theorem 1.1. LetM be a simply connected Riemannian symmetric space. Condition A holds unless M is the product of R and another symmetric space, in which case Condition A fails. Put another way, Condition A holds as long as the dimension of the center of G does not exceed 1. This is discussed in detail in section 4. Since the Levi-Civita connection on the tangent bundle of a symmetric space M is parallel, any bundle built from the tangent bundle will also admit a parallel connection, naturally induced from the Levi-Civita connection on TM . In particular k TM and Λ (TM) admit parallel connections, as does the (principal) bundle Fr(TM) of orthonormal frames of TM . Sub-bundles of Fr(TM), and finite covers of Fr(TM) or of sub-bundles of Fr(TM), also inherit parallel connections, whenever such subbundles are invariant by parallel transport. If B is a principal H-bundle over M and ρ : H → SO(k) is a representation of H, then the associated vector bundle E := B×ρR is the quotient of B×R by the PARALLEL CONNECTIONS OVER SYMMETRIC SPACES 3 equivalence (bh, v) ∼ (b, ρ(h)v). The equivalence class of (b, v) is denoted [(b, v)]. The vector bundle E inherits a connection from that of B, which will be parallel if the connection on B is parallel. Thus vector bundles with parallel connections naturally arise from representations of the holonomy group of a symmetric space M , or of its finite covers. Parallel connections are also related to group actions. Let G be a group that acts orthogonally and transitively on a vector bundle E, covering a group of isometries of M , and suppose that E has a connection that is invariant under this G-action. Then that connection is known to be parallel. Furthermore, for any fixed bundle E and G-action, Wang’s theorem [W] classifies the G-invariant connections. In this paper we show that, on symmetric spaces satisfying Condition A, these two constructions are equivalent, and are the only source of parallel connections: Main Theorem. Let M = G/H be a simply connected symmetric space written as a canonical group quotient, with G simply connected, and suppose that the canonical metric on M satisfies Condition A. Let E be any rank-k real vector bundle with parallel connection over M . Then there exists a representation ρ : H → SO(k) such that E is isomorphic to the vector bundle E′ = G×ρR, with the isomorphism taking the natural connection on E′ to the given connection on E. In [L], Laquer classified left invariant affine connections on compact irreducible Riemannian symmetric spaces. In the case of Riemannian connections, this is a special case of our Main Theorem when E is the tangent bundle of M , since as we will see in section 4, all irreducible symmetric spaces satisfy condition A. Corollary 1. Every parallel connection over M is invariant under a G-action on the bundle that covers the natural G-action on M . Corollary 2. Every unitary rank-k vector bundle with parallel connection over M is isomorphic to G×ρ′ C, where ρ′ : H → U(k) is a rank-k unitary representation of H. Several authors have investigated the question of when the action of a given transitive group of isometries on the base M can be lifted to a bundle over M , see for instance [B-H]. Since such a lifted action implies the existence of a parallel connection, Corollary 1 provides a complete answer: Over a symmetric space satisfying the hypotheses of the main theorem, the vector bundles that admit lifts of transitive group actions are exactly the bundles that admit parallel connections, which are exactly the associated vector bundles of G itself. Notice, though, that G is usually a cover of the isometry group G0 of the base, and that the action of the latter will not, in general, lift to the bundle: For example, the action of SO(5) on the 4-sphere does not lift to the spinor bundle, even though that of Spin(5) does. The main theorem tells us that a local condition on a bundle with connection, namely that its curvature be parallel, implies a rigid global structure. However, for this to be true we need assumptions about the underlying manifold, namely that it be a simply connected symmetric space whose metric connection satisfies Condition A. Without these conditions the conclusions of the theorem can be false, as the following counterexamples show. Let M = T 2 = G/H, where G = R and H = Z. M is not simply connected and Condition A fails, as kerR is all of Λ(Mp) while ImR is zero. Since 4 LUIS GUIJARRO, LORENZO SADUN, AND GERARD WALSCHAP R = 0, any bundle constructed from a representation of H is necessarily flat (although possibly with nontrivial holonomy). However, there exist bundles and connections over T 2 with parallel nonzero curvature. For example, consider the trivial complex line bundle over R. Taking the quotient of this by the relation f(x + n, y + m) = exp(2πiny)f(x, y) gives a complex line bundle of Chern class +1 over T 2 that we denote E. E admits many connections of constant nonzero curvature; one has connection form A = −2πixdy. Not only does this line bundle not come from a representation of H, but the connection is not invariant under translation. Indeed, the curvature itself measures the extent to which holonomy changes when a homologically nontrivial loop is translated. There is a 2-parameter family of parallel connections on E, indexed by T 2 itself, and translation takes one such connection into another. Now let M = G = R, with H trivial. M is simply connected but does not satisfy Condition A. Once again, any bundle built from the tangent bundle with its canonical structure has a flat connection, but there are connections over R that are parallel but not flat. For example, one can again look at a complex line bundle with connection form A = −2πixdy and constant curvature −2πidx ∧ dy. Unlike the T 2 example, this connection is invariant under translation, so Wang’s theorem does apply. The proof of the main theorem is based on the following observations. Let M = G/H be a simply connected symmetric space, with G simply connected. H is a covering space of Hol(M), the holonomy group ofM . Let π : H → Hol(M) be the covering map, so that TM = G×π R. Suppose E = G×ρR, for a representation ρ : H → SO(k), and endow E with the connection induced by the one on G. Then the holonomy around a loop γ for the three bundles TM , G, and E are related by the commutative diagram