Bochner-Yano Type Theorems for Conformal Killing Vector Fields under Curvature Pinching Conditions

Abstract

In this article, we investigate conformal Killing vector fields on closed Riemannian manifolds under a curvature pinching condition. By establishing a new Bochner-type identity for the $1$-form dual to a conformal Killing vector field, we derive a sharp gradient estimate via a Moser iteration procedure. Based on this estimate, we prove that, under a suitable upper bound on the Ricci curvature, every nontrivial conformal Killing vector field must be nowhere vanishing. Consequently, on even-dimensional manifolds with non-zero Euler characteristic, every conformal Killing vector field vanishes identically, which in turn implies that the conformal transformation group of such a manifold is finite. Our results extend the classical rigidity theorems of Yano and Bochner from the setting of non-positive Ricci curvature to that of small positive Ricci curvature, and generalize the recent results of Chen and Han from Killing vector fields to conformal Killing vector fields.