The aim of this work is to obtain new analitical solutions for Einstein equations in the anisotropical domain. This will be done via the minimal geometric deformation (MGD) approach, that allow us to decouple the Einstein equations. It requires a perfect fluid known solution that we will choose to be Finch‐Skeas(FS) solution. Two different constraints were applied, and in each case we found an interval of values for the free parameters, where necesarly other physical solutions shall live.

We describe a compactified Supermembrane, or M2-brane, with 2-form fluxes generated by constant three-forms that are turned on a 2-torus of the target space M 9 × T 2 . We compare this theory with the one describing a 11D M2-brane formulated on M 9 ×T 2 target space subject to an irreducible wrapping condition. We show that the flux generated by the bosonic 3-form under consideration is in a one to one correspondence to the irreducible wrapping condition. After a canonical transformation both Hamiltonians are exactly the same up to a constant shift in one particular case. Consequently both of them, share the same spectral properties. We conclude that the Hamiltonian of the M2-brane with 2-form target space fluxes on a torus has a purely discrete spectrum with eigenvalues of finite multiplicity and it can be considered to describe a new sector of the microscopic degrees of freedom of M-theory. We also show that the total membrane momentum in the direction associated to the flux condition adquires a quantized contribution in correspondence to the flux units that have been turned on.

We discuss the effect that small fluctuations of local anisotropy of pressure, and of energy density, may have on the occurrence of cracking in spherical compact objects, satisfying a polytropic equation of state. Two different kinds of polytropes are considered. For both, it is shown that departures from equilibrium may lead to the appearance of cracking, for a wide range of values of the parameters defining the polytrope. Prospective applications of the obtained results, to some astrophysical scenarios, are pointed out.

We study the Randall-Sundrum gravity under the gravitational decoupling through the minimal geometric deformation approach (MGD-decoupling). We show a family of new black hole solutions as well as new exact interior solutions for self-gravitating stellar systems and we discuss the corresponding matching conditions.

The main objective of this work, is to show two inequivalent methods to obtain new spherical symmetric solutions of Einstein's Equations with anisotropy in the pressures in isotropic coordinates. This was done inspired by the MGD method, which is known to be valid for line element in Schwarzschild coordinates. As example, we obtained two new analytical and physically acceptable solutions with each algorithm, using as seed solutions the known isotropic Gold III and Nariai IV solutions.