Uniqueness of low-dimensional generalized Ricci solitons

Establish that the known nontrivial generalized Ricci solitons on S^3 are the only nontrivial three-dimensional generalized Ricci solitons, and that their products with S^1 are the only four-dimensional generalized Ricci solitons.

Background

The paper proves that every nontrivial compact three-dimensional generalized Ricci soliton is diffeomorphic to a spherical space form S3/Γ, while every nontrivial compact four-dimensional generalized Ricci soliton is a finite quotient of S3 × S1. It also notes the existence of a nontrivial family of generalized Ricci solitons on S3.

These results provide topological restrictions but do not classify the soliton geometries themselves. The authors therefore formulate the conjecture that the known S3 family exhausts the three-dimensional case and that taking products with S1 exhausts the four-dimensional case.

References

There exists a nontrivial family of GRS on $S3$ (cf. Corollary 1.4). Theorem \ref{t:lowdGRStopology} lends some credence to the conjecture that these are the only such solitons, and that their products with $S1$ are the only four-dimensional GRS.

— Topology of low-dimensional generalized Ricci solitons and string backgrounds  (2608.25829 - Streets, 26 Aug 2026) in Introduction, Remark immediately following the proof of Theorem 1 (Theorem \ref{t:lowdGRStopology})