A review of the new of the problem of dark energy using modified gravity approach is considered. An explanation of the difficulties facing modern cosmology is given and different approaches are presented. We show why some models of gravity may suffer
of instabilities and how some are inconsistent with observations.
We show that the unification of electromagnetism and gravity into a single geometrical entity can be beautifully accomplished in a theory with non-symmetric affine connection (${Gamma}_{mu u}^{lambda} eq{Gamma}_{ umu}^{lambda}$), and the unifying sym
metry being projective symmetry. In addition, we show that in a purely-affine theory where there are no constrains on the symmetry of ${Gamma}_{mu u}^{lambda}$, the electromagnetic field can be interpreted as the field that preserves projective-invariance. The matter Lagrangian breaks the projective-invariance, generating classical relativistic gravity and quantum electromagnetism. We notice that, if we associate the electromagnetic field tensor with the second Ricci tensor and ${Gamma}_{[mu u]}^{ u}$ with the vector potential, then the classical Einstein-Maxwell equation can be obtained. In addition, we explain the geometrical interpretation of projective transformations. Finally, we discuss the importance of the role of projective-invariance in f(R) gravity theories.
A generalization to the Gibbons-Hawking-York boundary term for metric $f(R)$ gravity theories is introduced. A redefinition of the Gibbons-Hawking-York term is proposed. The proposed new definition is used to derive a consistent set of field equation
s and is extended to metric $f(R)$ gravity theories. The surface terms in the action are gathered into a total variation of some quantity. A total divergence term is added to the action to cancel these terms. Finally, the new definition is proven to demand no restrictions on the value of ${delta g}_{ab}$ or ${partial}_{c}{delta g}_{ab}$ on the boundary.
