Deformation of the $$\sigma _2$$σ2-curvature

Abstract

Our main goal in this work is to deal with results concern to the \(\sigma _2\)-curvature. First we find a symmetric 2-tensor canonically associated to the \(\sigma _2\)-curvature and we present an almost-Schur-type lemma. Using this tensor, we introduce the notion of \(\sigma _2\)-singular space and under a certain hypothesis we prove a rigidity result. Also we deal with the relations between flat metrics and \(\sigma _2\)-curvature. With a suitable condition on the \(\sigma _2\)-curvature, we show that a metric has to be flat if it is close to a flat metric. We conclude this paper by proving that the three-dimensional torus does not admit a metric with constant scalar curvature and nonnegative \(\sigma _2\)-curvature unless it is flat.