According to the “soul theorem” of J. Cheeger and D. Gromoll [Ann. Math. (2) 96, 413-443 (1972; Zbl 0246.53049)], a complete open manifold of nonnegative sectional curvature is diffeomorphic to the total space of the normal bundle of a compact totally geodesic submanifold, called the soul. In connection with this, a natural problem is to what extent the converse to the soul theorem holds. In other words, one asks which vector bundles admit complete nonnegative curved metrics. Various aspects of this problem have been studied by J. Cheeger [J. Differ. Geom. 8, 623-628 (1973; Zbl 0281.53040)], A. Rigas [J. Differ. Geom. 13, 527-545 (1978; Zbl 0441.55013)], M. Özaydin and G. Walschap [Proc. Am. Math. Soc. 120, 565-567 (1994; Zbl 0790.53038)], D. Yang [Pac. J. Math. 171, 569-583 (1995; Zbl 0859.53051)], K. Grove and W. Ziller [Ann. Math. (2) 152, 331-367 (2000; Zbl 0991.53016)] and the authors of the present paper [Math. Ann. 320, 167-190 (2001; Zbl 0997.53027)].
Now, in this paper the authors only deal with bundles over closed manifolds diffeomorphic to \(C\times T\), where \(C\) is simply-connnected and \(T\) is a standard torus. From J. Cheeger and D. Gromoll [op. cit.] any soul has a finite cover of this form, where \(\text{sec} (C)\geq 0\). Until recently, obstructions to the existence of metrics with \(\text{sec} \geq 0\) on vector bundles were only known for flats souls (cf. M. Özaydin and G. Walschap [op. cit.]), which corresponds to the case when \(C\) is a point. In their paper cited above the authors produced a variety of examples of vector bundles which do not admit complete metrics with \(\text{sec}\geq 0\). In all these examples \(\dim(T)>0\), in fact no obstructions are known to the existence of complete metrics with \(\text{sec}\geq 0\) on vector bundles over simply-connected nonnegatively curved manifolds.
Therefore, the main goal of this paper is to find obstructions in the case \(\dim(T)>0\). The main geometric ingredient is a splitting theorem proved formerly by the authors [op. cit.] and which says that after passing to a finite cover, the normal bundle to the soul can be taken, by a base-preserving diffeomorphism, to be the product \(\zeta_C\times T\) of a vector bundle \(\zeta_C\) over \(C\) with the torus. Then one needs to study the orbit of \(\zeta_C\times T\) under the action of the diffeomorphism group of \(C\times T\). Since vector bundles are rationally classified by the Euler and Pontryagin classes, the problem reduces to analyzing the action of \(\text{Diffeo}(C\times T)\) on the rational cohomology algebra \(H^* (C\times T,\mathbb{Q})\) of \(C\times T\).
The main results of the paper are stated by using the following technical definition. Given a vector bundle \(\zeta\) over \(C\times T\), one says that \(\zeta\) virtually comes from \(C\) if for some finite cover \(p:T\to T\), the pullback of \(\zeta\) by \(\text{id}_C\times p\) is isomorphic to the product \(\zeta_C\times T\) where \(\zeta_C\) is a bundle over \(C\). Then the following theorems are proved:
Theorem 1.1. Let \(C\) be a closed smooth simply-connected manifold, and let \(T\) be a torus. Let \(\zeta\) be a rank two vector bundie over \(C\times T\). If \(E(\zeta)\) admits a complete metric with \(\text{sec}\geq 0\), then \(\zeta\) virtually comes from \(C\).
Theorem 1.2. Let \(C=G//H\) be a simply-connected biquotient of compact Lie groups such that \(H\) is semisimple, and let \(T\) be a torus. Let \(\zeta\) be a vector bundle over \(C\times T\) of \(\text{rank}\leq 4\). If \(E(\zeta)\) admits a complete metric with \(\text{sec}\geq 0\), then \(\zeta\) virtually comes from \(C\).
Let \({\mathcal H}\) be the class of simply-connected finite CW-complexes whose rational cohomology algebras have no zero derivations of negative degree.
Theorem 1.3. Let \(C\in {\mathcal H}\) be a closed smooth manifold, and let \(T\) be a torus. If \(\zeta\) is a vector bundle over \(C\times T\) such that \(E(\zeta)\) admits a complete metric with \(\text{sec}\geq 0\), then \(\zeta\) virtually come from \(C\).
The class \({\mathcal H}\) contains any compact simply-connected Kähler manifold and any compact homogeneous space \(G/H\) such that \(G\) is a compact connected Lie group and \({\mathcal H}\) is a closed subgroup with \(\text{rank}(H)= \text{rank}(G)\). Also the total space of a fibration belongs to \({\mathcal H}\) provided the base and the fiber do also.
S. Halperin conjectured that any elliptic space \(C\) of positive Euler characteristic belongs to \({\mathcal H}\). This conjecture, which is considered one of the central problems in rational homotopy theory, has been confirmed in several important cases. Then, K. Grove and S. Halperin [Publ. Math., Inst. Hautes Etud. Sci. 56, 171-177 (1982; Zbl 0508.55013)] have conjectured that any closed simply connected nonnegatively curved manifold is elliptic. Finally, the authors note that, if the above conjectures are true, then \({\mathcal H}\) contains any simply-connected compact nonnegatively curved manifold of positive Euler characteristic.
Theorem 1.4. Let \(C=SU(6)/(SU(3) \times SU(3))\) and \(\dim(T)\geq 2\). Then there exists a rank six vector bundle \(\zeta\) over \(C\times T\) which does not virtually come from \(C\), but \(E(\zeta)\) admits a complete metric of \(\text{sec}\geq 0\) such that the zero section is a soul.
The paper contains also some applications and examples and several open problems.