Two-dimensional Palatini Gauss-Bonnet gravity on a strip reduces to boundary geodesics on the SL(2,R) group manifold.
This constraint implies that the theory has no degree of freedom in the absence of boudaries, even if the spatial topology is non-trivial (circle)
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Exact solution of two-dimensional Palatini Gauss-Bonnet theory on a strip
Two-dimensional Palatini Gauss-Bonnet gravity on a strip reduces to boundary geodesics on the SL(2,R) group manifold.