Mimetic gravity in the extended objects framework

Starting from the most general second-order in derivatives theories describing extended objects of arbitrary dimension evolving geodetically in a codimension-one flat ambient space-time, we determine the subset of models yielding second-order equations of motion, forming an intriguing theory known as Lovelock-type brane gravity (LBG). These models further extend the so-called geodetic brane gravity (GBG) approach, thereby naturally promoting the GBG geometric properties, allowing LBG to be reformulated as a mimetic embedding gravity and, in turn, the possibility of introducing fictional matter through a peculiar current $\mathcal{T}^{a\,\mu}$. Grounded in the elasticity theory, we provide a possible origin of such a current. Finally, variational techniques are employed to elucidate the mechanical function of both the dark current $\mathcal{T}^{a\,\mu}$ and its tangential components $\mathcal{T}^{ab}$; these serve as the constituents of a fictional energy-momentum tensor that shares characteristics with a perfect fluid.