Deformation of the σ₂-curvature

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Our main goal in this work is to deal with results concern to the σ₂-curvature. First we find a symmetric 2-tensor canonically associated to the σ₂-curvature and we present an Almost Schur Type Lemma. Using this tensor we introduce the notion of σ₂-singular space and under a certain hypothesis we prove a rigidity result. Also we deal with the relations between flat metrics and σ₂-curvature. With a suitable condition on the σ₂-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 σ₂-curvature unless it is flat.