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We propose a procedure for the $D\rightarrow 4$ limit ofEinstein-Gauss-Bonnet (EGB) gravity that leads to a well defined actionprinciple in four dimensions. Our construction is based on compactifying$D$-dimensional EGB gravity on a $(D-4)$-dimensional maximally symmetric spacefollowed by redefining the Gauss-Bonnet coupling $\alpha\rightarrow\frac{\alpha}{D-4}$. The resulting model is a special scalar-tensor theory thatbelongs to the family of Horndeski gravity. Static black hole solutions in thescalar-tensor theory are investigated. Interestingly, the metric profile isindependent of the curvature of the internal space and coincides with the$D\rightarrow 4$ limit of the usual EGB black hole with the unusualGauss-Bonnet coupling $\frac{\alpha}{D-4}$. The curvature information of theinternal space is instead encoded in the profile of the extra scalar field. Ourprocedure can also be generalized to define further limits of the Gauss-Bonnetcombination by compactifying the $D$-dimensional theory on a$(D-p)$-dimensional maximally symmetric space with $p\leq 3$. These lead todifferent $D\rightarrow 4$ limits of EGB gravity as well as its $D\rightarrow2,3$ limits.

Horndeski Gravity as $D\rightarrow4$ Limit of Gauss-Bonnet