Deformation of the $\sigma_2$-curvature

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 3-dimensional torus does not admit a metric with constant scalar curvature and non-negative $\sigma_2$-curvature unless it is flat.