Experimental evidence suggests that we live in a spatially flat, accelerating universe composed of roughly one-third of matter (baryonic + dark) and two-thirds of a negative-pressure dark component, generically called dark energy. The presence of such energy not only explains the observed accelerating expansion of the Universe but also provides the remaining piece of information connecting the inflationary flatness prediction with astronomical observations. However, despite of its good observational indications, the nature of the dark energy still remains an open question. In this paper we explore a geometrical explanation for such a component within the context of brane-world theory without mirror symmetry, leading to a geometrical interpretation for dark energy as warp in the universe given by the extrinsic curvature. In particular, we study the phenomenological implications of the extrinsic curvature of a Friedman-Robertson-Walker universe in a five-dimensional constant curvature bulk, with signatures (4,1) or (3,2), as compared with the X-matter (XCDM) model. From the analysis of the geometrically modified Friedman's equations, the deceleration parameter and the Weak Energy Condition, we find a consistent agreement with the presently known observational data on inflation for the deSitter bulk, but not for the anti-deSitter case.

The concept of smooth deformations of a Riemannian manifolds, recently evidenced by the solution of the Poincaré conjecture, is applied to Einstein's gravitational theory and in particular to the standard FLRW cosmology. We present a brief review of the deformation of Riemannian geometry, showing how such deformations can be derived from the Einstein-Hilbert dynamical principle. We show that such deformations of space-times of general relativity produce observable effects that can be measured by four-dimensional observers. In the case of the FLRW cosmology, one such observable effect is shown to be consistent with the accelerated expansion of the universe.

The most general geometrical scenario in which the brane-world program can be implemented is investigated. The basic requirement is that it should be consistent with the confinement of gauge interaction, the existence of quantum states and the embedding in a bulk with arbitrary dimensions, signature and topology. It is found that the embedding equations are compatible with a wide class of Lagrangians, starting with a modified Einstein-Hilbert Lagrangian as the simplest one, provided minimal boundaries are added to the bulk. A non-trivial canonical structure is derived, suggesting a canonical quantization of the brane-world geometry relative to the extra dimensions, where the quantum states are set in correspondence with high frequency gravitational waves. It is shown that in the cases of at least six dimensions, there exists a confined gauge field included in the embedding structure. The size of extra dimensions compatible with the embedding is calculated and found to be different from the one derived with product topology.

The standard Friedmann universe embedded in a five dimensional and constant curvature bulk is examined without any a priori junction condition between the brane and the bulk. A geometrical explanation for the accelerated expansion of the universe is derived by using a minimum set of assumptions consistent with the brane-world program. It is shown that the extrinsic curvature of the brane can be associated to the dark energy which presumably drives the universe expansion.Comment: Revtex, 4 page