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In the Riemann geometry, the metric's equation of motion derived from an arbitrary Lagrangian can be succinctly expressed in term of the first variation of the action with respect to the Riemann tensor if it were independent of the metric. We refer to this variation as the E-tensor.
Only for specific Lagrangians the E-tensor and the metric's equation of motion might be of the same differential degree. Requiring the same differential degree for the E-tensor and the metric's equation, therefore, is a criterion to restrict the form of Lagrangians, or the corrections to the Einstein-Hilbert action. For actions in form of general functional of the metric and Riemann tensor, just Lovelock gravity meets this criterion. We consider functional of the first covariant derivative of the Riemann tensor and metric. We find a family of the Lagrangians that meets the criterion, a family that leads to fourth order differential equation for the metric.
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