TRANSITIVE HOLONOMY GROUP AND

RIGIDITY IN NONNEGATIVE CURVATURE

Luis Guijarro and Gerard Walschap

Abstract. In this note, we examine the relationship between the twisting of a vector bundle ξ over a manifold M and the action of the holonomy group of a Riemannian connection on ξ. For example, if there is a holonomy group which does not act transitively on each fiber of the corresponding unit sphere bundle, then for any f : S^{n} → M, the pullback f^{∗}ξ of ξ admits a nowhere-zero cross section. These facts are then used to derive a rigidity result for complete metrics of nonnegative sectional curvature on noncompact manifolds.

Let ξ denote a rank k vector bundle over a simply connected manifold M^{n}. If ξ admits a Riemannian connection with trivial holonomy group, then ξ is a trivial bundle. On the other hand, it is easy to construct examples of trivial bundles with transitive holonomy (in the sense that the holonomy group acts transitively on each fiber of the unit sphere bundle of ξ). A natural question to ask is how twisted must the bundle be in order to guarantee that every connection will have transitive holonomy. Clearly, such a bundle cannot admit a nowhere-zero section. It turns out that when the base is a sphere, this necessary condition is also sufficient (see Example 2). For arbitrary base, this no longer holds, but a slightly stronger condition is sufficient: Let ξ^{1} denote the unit sphere bundle of ξ; if ξ^{1} admits a section, then for q ≥ 2, one has a short exact sequence

(1)  0 ⟶ π_{q}(S^{k−1}) ⟶^{ı_{∗}} π_{q}(E^{1}) ⟶^{π_{∗}} π_{q}(M) ⟶ 0

which in fact splits. Here ı is the fiber inclusion, π the bundle projection, and E^{1} the total space of ξ^{1}. We show that if ξ admits a connection with nontransitive holonomy, then (1) holds for all q ≥ 2.

It easily follows that on bundles over spheres which admit no nonvanishing section, any connection has transitive holonomy. Other examples include the canonical line bundle over complex or quaternionic projective space P^{n}, the tangent bundle of P^{n} for even n, the canonical bundles over the Grassmannians G_{k}(ℝ^{n}) and G_{k}(ℂ^{n}) of k-planes in n-space for

Typeset by AMS-TEX

The first author was supported by a postdoctoral fellowship of the Comunidad Autónoma de Madrid.

large n, and consequently the universal bundles over the classifying spaces BSO(k) and BU(k). All these examples underscore the general idea that the holonomy group increases as the bundle becomes more twisted.

This property has an interesting application to the geometry of open manifolds N with nonnegative sectional curvature: recall that such a space contains a compact totally geodesic submanifold S, called a soul of N. Many of the bundles mentioned above admit such a metric. The above result implies that actually any complete metric of nonnegative curvature on these bundles is quite rigid:

Theorem. Let N denote a complete manifold with nonnegative sectional curvature and soul S. If the normal bundle ν of S is one of those mentioned above, i.e:

(1) The unit sphere bundle of ν is a homotopy sphere,

(2) S is a sphere and ν does not admit a nowhere-zero section,

(3) S = ℂℙ^{n} or ℍℙ^{n} with n even, and ν is the tangent bundle,

(4) S = G_{k}(ℝ^{n}) with n ≥ 2k, and ν is the canonical bundle, then the exponential map exp : ν → N of the normal bundle of S in M is a diffeomorphism, the metric projection N → S onto the soul is a C^{∞} Riemannian submersion, and the ideal boundary N(∞) of M consists of a single point.

1. Nontransitive holonomy

Theorem. Let E denote the total space of an oriented Euclidean rank k bundle over a manifold M. If this bundle admits a Riemannian connection whose holonomy does not act transitively on the unit sphere bundle E^{1} of E, then for q ≥ 2, there is a short exact sequence

(1)  0 ⟶ π_{q}(S^{k−1}) ⟶^{ı_{∗}} π_{q}(E^{1}) ⟶^{π_{∗}} π_{q}(M) ⟶ 0.

Proof. Denote by B the principal SO(k) bundle of E^{1}, and consider a connection on E with nontransitive holonomy. Let G denote the holonomy group of the connection, and P the holonomy bundle through some b ∈ B; i.e., P consists of all points of B that can be joined to b by some horizontal curve. P is a principal subbundle of B [P], and E^{1} is equivalent to P ×_{G} S^{k−1}. In other words, E^{1} is the base of the principal G-fibration

ρ : P × S^{k−1} → E^{1}, ρ(pg, u) = ρ(p, gu), p ∈ P, u ∈ S^{k−1}, g ∈ G.

Let us fix some u ∈ S^{k−1}, and define a map h : P → E^{1} by h(p) = ρ(p, u). Notice that if G_{p} denotes the fiber of P through p and F_{h(p)} the fiber of E^{1} through h(p), then h : (P, G_{p}) → (E_{1}, F_{h(p)}): In fact, if q = pg ∈ G_{p} for some g ∈ G, then h(q) = ρ(pg, u) = ρ(p, gu) ∈ F_{h(b)}.

Since the holonomy is not transitive, h : G_{p} → F_{h(p)} is not onto, and is then homotopically trivial because a sphere with a point deleted is contractible. Consider the commutative diagram

π_{l}(M) ⟵ π_{l}(P, G_{p}) ⟶^{∂} π_{l−1}(G_{p})
∥  ^{h_{∗}}↓  ↓^{h_{∗}}
π_{l}(M) ⟵ π_{l}(E^{1}, F_{h(p)}) ⟶_{∂} π_{l−1}(F_{h(p)})

where the unlabeled arrows are the isomorphisms induced by the respective bundle projections, and ∂ is the boundary operator, cf. e.g. [S]. Now the h_{∗} on the left is an isomorphism since it commutes with the bundle projections, whereas the h_{∗} on the right is trivial. Thus, the lower ∂ is trivial. The theorem follows by looking at the homotopy sequence of the fibration E^{1} → M. □

2. Applications

We now consider several bundles satisfying the hypotheses of the Theorem, so that any Riemannian connection on them must have transitive holonomy:

Example 1. A vector bundle whose unit sphere bundle is a homotopy-sphere. Indeed, if the base has dimension n, then π_{k−1}(E^{1}) = π_{k−1}(S^{n+k−1}) = 0, so that one cannot have a short exact sequence as in (1) in dimension k − 1. This result applies in particular to the canonical line bundles over complex and quaternionic projective spaces as well as to the vector bundle E over S^{8} whose unit bundle corresponds to the Hopf fibration S^{7} → S^{15} → S^{8} (see also Example 4 below for further related results).

Before discussing our next example, we point out the following consequence of the Theorem:

Corollary 1. Let ξ denote an oriented vector bundle over M. If ξ admits a connection with nontransitive holonomy, then for any map f : S^{n} → M from an n-sphere to M (n ∈ ℕ), the induced bundle f^{∗}ξ admits a nowhere-zero cross-section.

Proof. By the Theorem, p_{∗} : π_{n}(E^{1}) → π_{n}(M) is onto, where, as before, E_{1} is the total space of the unit sphere bundle of ξ. Thus, there exists α ∈ π_{n}(E_{1}) such that p_{∗}(α) = [f]. By the covering homotopy property, there exists a representative g : S^{n} → E_{1} of α such that p ∘ g = f; i.e., g induces a section of f^{∗}ξ.

Example 2. (see also [GW]) Let τ denote a vector bundle over a homotopy-sphere M. Then the following statements are equivalent:

(1) Any connection on τ has transitive holonomy (2) τ does not admit a nowhere-zero section

(3) τ does not split as a Whitney sum τ_{1} ⊕ τ_{2} of subbundles τ_{1}, τ_{2}.

Proof.

(1) ⇒ (3): If τ splits, then a connection on each subbundle induces a ‘direct sum’ connection on the bundle that leaves each factor invariant. The resulting holonomy group is clearly nontransitive.

(3) ⇒ (2): Trivial.

(2) ⇒ (1): This is an immediate consequence of Corollary 1 by taking f : S^{n} → M to be a homotopy equivalence.

Example 3. The tangent bundles of the projective spaces ℂP^{n} and ℍP^{n} for even n. The argument is quite similar in both cases, so we shall only outline the one for complex projective space: Let ρ : S^{2n+1} → ℂP^{n} denote the Hopf fibration, and consider the pullback ρ^{∗}τ of the tangent bundle of ℂP^{n} to the sphere. If τ admits a connection with nontransitive holonomy, then ρ^{∗}τ admits a nowhere-zero section by Corollary 1. But ρ^{∗}τ is canonically identified with a codimension 1 subbundle of T S^{2n+1}, and this would imply that S^{2n+1} admits two linearly independent vector fields. This is impossible for even n by [S, Appendix 7]. It should be noted that when n > 1 is odd, there do exist nontransitive connections on the tangent bundle of ℂP^{n} (this was pointed out by B. Wilking): The reason is that the Hopf fibration S^{4k+3} → ℍP^{k} factors through ℂP^{2k+1}, yielding a fibration S^{2} → ℂP^{2k+1} → ℍP^{k}; the tangent bundle of ℂP^{2k+1} therefore splits as a Whitney sum, and any connection which leaves each factor invariant will have nontransitive holonomy.

Example 4. Let γ_{k}^{n} denote the canonical bundle over the Grassmannian G_{k}(ℝ^{n}) of oriented k-planes in ℝ^{n}. Thus the fiber of γ_{k}^{n} over a k-dimensional subspace is the subspace itself. We claim that if n is large enough, then any connection on γ_{k}^{n} has transitive holonomy: To see this, it suffices to show that for any integer k, there exists a rank k bundle over some sphere which does not admit a nowhere-zero cross-section; indeed, such a bundle is the pullback of γ_{k}^{2k} for some map from the base into G_{k}(ℝ^{2k}) (see e.g. [M]), so if the latter were to admit a nontransitive connection, then the former would admit a nowhere-zero cross-section by Corollary 1. Now, for even k, one can just take the tangent bundle of S^{k}. For odd k, we argue as follows: Recall that oriented rank k bundles over S^{l} are classified by elements of π_{l−1}SO(k). Since such a bundle admits a nonzero section iff. its structure group reduces to SO(k − 1), it suffices to show that the homomorphism ı_{∗} : π_{n}SO(k − 1) → π_{n}SO(k) induced by the inclusion ı is not onto for some value of n. Consider the portion of the long exact homotopy sequence

⋯ → π_{2k−3}SO(k − 1) →^{ı_{∗}} π_{2k−3}SO(k) → π_{2k−3}(S^{k−1}) →^{∂} π_{2k−4}SO(k − 1) → ⋯

of the orthogonal fibration. It is a standard fact that for odd k > 1, the third group contains an infinite cyclic subgroup (see e.g. [W p. 671]). Thus if the inclusion homomorphism were onto, then the boundary homomorphism ∂ : π_{2k−3}(S^{k−1}) → π_{2k−4}SO(k − 1) would be injective, and π_{2k−4}SO(k−1) would contain an infinite cyclic subgroup. This is impossible, since even-dimensional homotopy groups of compact semi-simple Lie groups are always finite by [Se, V3 Corollary 1].

Corollary 2. Any SO(k)-connection on the universal bundle γ_{k} over the classifying space BSO(k) has transitive holonomy.

Proof. If γ_{k} were to admit a connection with nontransitive holonomy, then the pullback of this connection to γ_{k}^{n} via the inclusion G_{k}(ℝ^{n}) ↪ BSO(k) would also have nontransitive holonomy by Example 4. Alternatively, one can directly apply the Theorem to the corresponding sphere bundle, since its total space is homotopy equivalent to BSO(k − 1) and the corresponding exact homotopy sequence is that of the orthogonal fibration, but shifted:

⋯ ⟶^{∂} π_{q}(S^{k−1}) ⟶^{ı_{∗}} π_{q}(BSO(k − 1)) ⟶^{π_{∗}} π_{q}(BSO(k)) ⟶ ⋯
∥  ∥  ∥
⋯ ⟶ π_{q}(S^{k−1}) ⟶^{∂} π_{q−1}SO(k − 1) ⟶^{ı_{∗}} π_{q−1}SO(k) ⟶^{π_{∗}} ⋯

A similar statement holds for U(k)-connections on the canonical bundles over complex Grassmannians (and on BU(k)), since the above argument can be applied to the fibration U(k) → U(k + 1) → S^{2n+1}.

Corollary 3. Let ξ denote an oriented vector bundle of rank k over M^{n}; suppose that M is (k − 1)-connected and π_{k}(M) is cyclic. If ξ admits a connection with nontransitive holonomy, then the Euler class of ξ is 0.

Remark. We are implicitely assuming that k ≤ n in the hypotheses of Corollary 3, since otherwise ξ admits a nowhere-zero section, and the conclusion is trivial. Notice also that the assumptions on M are satisfied if M is a homology k-sphere.

Proof. Let α denote a generator of π_{k}(M), and f : S^{k} → M a representative of α. Then f_{∗} : π_{i}(S^{k}) → π_{i}(M) is an isomorphism for i < k and an epimorphism for i = k. By a result of J.H.C. Whitehead (see e.g. [B, page 481]), the same is true for f_{∗} : H_{i}(S^{n}) → H_{i}(M). By the universal coefficient theorem, f^{∗} : H^{k}(M) → H^{k}(S^{n}) is a monomorphism. But then the Euler class e(ξ) must vanish, since e(f^{∗}(ξ)) = f^{∗}(e(ξ)), and f^{∗}(ξ) admits a nowhere-zero section by Corollary 2.

The bundles in Examples 1 and 3 all admit Riemannian connections with parallel curvature tensor, so that by [SW], they also admit complete metrics with nonnegative sectional curvature K and soul isometric to the zero section. It is still an open question [CG] whether every bundle over S^{n} carries such a metric. No counterexample has so far been found, and in fact, when n ≤ 4, every bundle admits a metric with K ≥ 0. In any case, for those that do, the metric must be quite rigid if the bundle is twisted enough:

Corollary 4. Let N denote a complete manifold with nonnegative sectional curvature and soul S. If the normal bundle ν(S) is one of those listed in the above examples, then:

(1) The exponential map exp : ν(S) → N is a diffeomorphism.

(2) The metric projection N → S is a C^{∞} Riemannian submersion.

(3) The ideal boundary N(∞) consists of a single point.

Proof. Given u ∈ ν(S), |u| = 1, let γ_{u} denote the geodesic in direction u. (1) is an immediate consequence of the fact that if γ_{u} is a ray (i.e., a geodesic with no conjugate points in [0, ∞)), then γ_{v} is again a ray for any parallel translate v of u [W]. For (2), it is known that the metric projection N → S is a C^{2} Riemannian submersion [G]. It follows from (1) that this projection equals π_{ν} ∘(exp^{−1}), and is therefore C^{∞}. Finally, to establish (3), it suffices to show that any 2 rays γ_{u} and γ_{v} remain at bounded distance from each other. But if c is a curve that parallel translates u to v, one obtains a flat totally geodesic rectangle by exponentiating the parallel section along c [CG]. Thus, the distance between γ_{u}(t) and γ_{v}(t) is always bounded above by the length of c.

As a concluding remark, we point out once again that if the base of the bundle is not a sphere, then nontransitive holonomy does not, in general, imply the existence of a nowhere-zero section. In fact, even the weaker assertion that nontransitive holonomy implies vanishing of the Euler class is not true (unless, of course, some additional assumptions such as those in Corollary 3 hold):

Example 5. Let M^{3} be a compact oriented flat 3-manifold with H^{2}(M; ℤ) = ℤ_{4} × ℤ_{4} (see e.g. [Wo, page 122]). Any α ∈ H^{2}(M) represents the integral Euler class of a unique oriented plane bundle over M. Consider a plane bundle over M whose integral Euler class is non-zero. Since the rational Euler class vanishes, the bundle admits a flat connection by [K-T]. Thus the reduced holonomy group is trivial, and the holonomy group itself does not act transitively on the unit sphere bundle.

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Department of Mathematics, Universidad Autonoma de Madrid, 28049 Madrid, Spain Current address: Department of Geometry and Topology, Facultad de Ciencias Matemáticas, Univer-

sidad Complutense de Madrid, Madrid 28040, Spain

   E-mail address: luis.guijarro@mat.ucm.es

Department of Mathematics, University of Oklahoma, Norman OK 73019

   E-mail address: gerard@aftermath.math.ou.edu