Kirby Problem 2.23

Definitions and mathematical statement

Let $B$ and $F$ be closed connected oriented surfaces of genera at least two. A smooth surface bundle over a surface is a smooth locally trivial map $p:E\to B$ with fiber $F$ and orientation-preserving transition maps. Its total space $E$ is a closed oriented smooth $4$-manifold. A real hyperbolic metric on $E$ is a Riemannian metric with sectional curvature identically $-1$. A metric of strictly negative curvature has $K_E(\sigma)<0$ for every two-dimensional tangent plane $\sigma$; the curvature need not be constant. On a closed manifold these metrics are complete.

Does there exist such a bundle whose total space admits a real hyperbolic metric? More generally, does one exist whose total space admits any metric of strictly negative sectional curvature? The genera, bundle, and metric may all be chosen freely. The bundle projection need not be a Riemannian submersion, and the fibers need not be totally geodesic. Thus the metric condition is on the total four-dimensional manifold, not merely on its surface fibers and base. The constant-curvature existence question implies the variable-curvature one.

Short English statement

Can a closed surface bundle over a closed surface carry a real hyperbolic metric, or any negatively curved metric?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2771 (KP-2.23), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 2.23. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.