ar X iv : d g - ga / 9 71 20 14 v 1 2 2 D ec 1 99 7 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.