Isometric Splitting of Metrics Without Conjugate Points on $\Sigma \times S^1$

Abstract

We establish a global rigidity theorem for Riemannian metrics without conjugate points on three-manifolds of the form $M = \Sigma \times S^1$, where $\Sigma$ is a compact orientable surface of genus at least 2. The main result states that any such metric must be a Riemannian product, with universal cover isometric to $(\mathbb{H}^2, g_0) \times (\mathbb{R}, dt^2)$. This extends the classical Hopf conjecture from tori to this natural class of manifolds with non-abelian fundamental group containing a central $\mathbb{Z}$ factor. We provide two independent proofs: one utilizing the regularity of Busemann functions and stability theory of the Riccati equation along Killing flows, and another based on a detailed analysis of the curvature operator acting on Jacobi fields. We derive sharp geometric inequalities, analyze the deformation space of such metrics, and discuss several geometric and dynamical consequences, including marked length spectrum rigidity, constraints on topological entropy, and incompatibility with non-trivial Sasakian structures.