  We present the perturbation equations in an embedded four space-time from the
linear Nash-Greene fluctuations of background metric. In the context of a
five-dimensional bulk, we show that the cosmological perturbations are only
propagated by the gravitational tensorial field equation. In Newtonian
conformal gauge, we study the matter density evolution in sub-horizon regime
and on how such scale may be affected by the extrinsic curvature. We apply a
joint likelihood analysis to the data by means of the Markov Chain Monte Carlo
(MCMC) method for parameter estimation using a pack of recent datasets as the
Pantheon Supernovae type Ia, the Baryon Acoustic Oscillations (BAO) from DR12
galaxy sample and Dark Energy Survey (DESY$1$). We discuss the tensions on the
Hubble constant $H_0$ and the growth amplitude factor $S_8$ of the observations
from Planck 2018 Cosmic Microwave Background (CMB) and the local measurements
of $H_0$ with Hubble Space Telescope (HST) photometry and Gaia EDR3. As a
result, we obtain an alleviation below $\sim 1\sigma$ in the contours
($S_8$-$\Omega_m$) at $68.4\%$ confidence level (CL) when compared with DESY1
data. On the other hand, the $H_0$ tension persists with $\sim 2.6\sigma$ at
$68.4\%$ CL and $1.95\sigma$ at $95.7\%$ CL, aggravated with the inclusion of
BAO data.
