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The gravitational Lagrangian can be written as a summation of a bulk and atotal derivative term. For some theories of gravity such as Einstein gravity,or more general Lovelock gravities, there are Lagrangian holographic relationsbetween the bulk and the total derivative term such that the latter is fullydetermined by the former. However at the $D\rightarrow 2\&4$ limit, the bulksof Einstein or Gauss-Bonnet theories become themselves total derivatives.Performing the Kaluza-Klein reduction on Einstein and Gauss-Bonnet gravitiesgives rise to some two-dimensional or four-dimensional scalar-tensor theoriesrespectively. We obtain the holographic relations for the $D = 2$ and $D = 4$cases, which have the same form as the holographic relations in pure gravity inthe foliation independent formalism.

On the Lagrangian Holographic Relation at $D\rightarrow2$ and $4$ Limits of Gravity