Complete shrinking Riemann solitons

Purpose This paper investigates the topological and geometric properties of complete shrinking Riemann solitons (Mm, g, µ, V), extending classical results from Ricci solitons to the more general Riemann soliton setting. Design/methodology/approach We employ techniques from Riemannian geometry, including the analysis of the Riemann curvature tensor, Lie derivatives along vector fields and comparison theorems for geodesics. By establishing suitable inequalities on the divergence and norm of the soliton vector field V, we derive diameter bounds and compactness criteria. Findings We prove that any complete shrinking Riemann soliton is compact if the divergence of V satisfies divV ≤ −K1 and ||V || ≤ K2 for positive constants K1 and K2. Moreover, we show that the fundamental group of such a manifold is finite. Explicit diameter estimates in terms of K1, K2 and the soliton constant µ are provided. Originality/value These results generalize known compactness and finiteness theorems for Ricci solitons to the framework of Riemann solitons, offering new insights into their geometric structure and topological constraints.