  In the present paper we apply the Nash's theory of perturbative geometry to
the study of dark matter gravity in a higher-dimensional space-time. It is
shown that the dark matter gravitational perturbations at local scale can be
explained by the extrinsic curvature of the standard cosmology. In order to
test our model, we use a spherically symmetric metric embedded in a
five-dimensional bulk. As a result, considering a sample of 10 low surface
brightness and 6 high surface brightness galaxies, we find a very good
agreement with the observed rotation curves of smooth hybrid alpha-HI
measurements.
