Geometrical aspects on the dark matter problem

• The metric perturbation and the embedding lead naturally to a “brane-world”-like higher dimensional structure.
• Nash’s theorem as a cornerstone of the formation of geometrical structures.
• The dark matter gravitational perturbations at local scale can be explained by the extrinsic curvature.
• A good agreement was found with the observed rotation curves of smooth hybrid alpha-HI measurements.

In the present paper we apply 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.