Harmonic Fibrations of Exotic Complex Projective Space Bundles

Definitions and mathematical statement

Let $S^1$ denote the circle and $\mathbb{CP}^3$ complex projective three-space. The supplied formulation concerns a smooth manifold $M$ homeomorphic to $S^1\times\mathbb{CP}^3$, hence a closed connected smooth manifold of real dimension seven, belonging to the "exotic cases" of a preceding $S^1$-bundle problem. The rewritten source describes these cases as manifolds diffeomorphic to an exotic connected sum, but does not identify its summands or the precise smooth structures under consideration.

For a Riemannian metric $g$ on $M$, a smooth map $f:(M,g)\to S^1$, with the usual circle metric, is harmonic if its tension field vanishes: \[ \tau_g(f)=\operatorname{tr}_g(\nabla {\,\mathrm{d}} f)=0. \] Here a smooth fibration means a smooth locally trivial fiber bundle over $S^1$. For a smooth map from this closed manifold to $S^1$, this is equivalent to being a submersion, namely \[ {\,\mathrm{d}} f_x\ne 0\qquad\text{for every }x\in M. \]

Question, with unresolved source qualifications. In the exotic cases referred to by the source, can the harmonic map $f:(M,g)\to S^1$ be a smooth fibration?

The supplied record does not specify the preceding bundle construction, the exotic connected sums, the homotopy class selecting "the harmonic map," or the quantifier on $g$. It therefore does not determine whether the question asks for some metric admitting such a harmonic fibration, or concerns a metric already supplied by the preceding construction. These missing qualifications must be recovered before a fully quantified statement can be fixed. In particular, no homotopy class is imposed here: constant maps are harmonic and cannot be fibrations, so the choice of class materially affects the question.

Short English statement

For the exotic smooth seven-manifolds homeomorphic to $S^1\times\mathbb{CP}^3$ intended by the source, ask whether the associated harmonic map to the circle can be a smooth fiber bundle projection. The exotic cases, the map's homotopy class, and the metric quantifiers remain unspecified in the supplied record.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30003375$, source problem code catalogue entry, title "Harmonic Fibrations of Exotic Complex Projective Space Bundles," collection "Oberwolfach Reports Open Problems." No individual proposer is identified; the supplied proposed year is 2017.
• Supplied source citation: Mini-Workshop: Spaces and Moduli Spaces of Riemannian Metrics (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/3, pp. 133--166. https://doi.org/10.4171/owr/2017/3. The citation's parenthetical year 2018 differs from the report identifier and proposed year 2017; this discrepancy is unresolved.
• Supplied literature source: "Oberwolfach Report 3/2017," https://ems.press/doi/pdf/10.4171/OWR/2017/3.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes it as a follow-up to the exotic smooth classification of the $S^1\times\mathbb{CP}^3$ type. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official passage was checked and rewritten as a self-contained problem. These are supplied verification claims; no independent source inspection is asserted in this draft.