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===Curvature=== The [[Ricci tensor]] is defined as the contraction of the [[Riemann tensor]]. Some authors use the contraction <math>R_{ab} \, = R^c{}_{acb}</math>, whereas others use the alternative <math>R_{ab} \, = R^c{}_{abc}</math>. Due to the [[Riemann tensor#Symmetries and identities|symmetries of the Riemann tensor]], these two definitions differ by a minus sign. In fact, the second definition of the Ricci tensor is <math>R_{ab} \, = {R_{acb}}^c</math>. The sign of the Ricci tensor does not change, because the two sign conventions concern the sign of the Riemann tensor. The second definition just compensates the sign, and it works together with the second definition of the Riemann tensor (see e.g. Barrett O'Neill's Semi-riemannian geometry).
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