We consider a non-relativistic (NR) limit of (2 + 1)-dimensional Maxwell Chern-Simons (CS) gravity with gauge algebra [Maxwell] ⊕ u(1) ⊕ u(1). We obtain a finite NR CS gravity with a degenerate invariant bilinear form. We find two ways out of this difficulty: To consider i) [Maxwell] ⊕ u(1), which does not contain Extended Bargmann gravity (EBG); or, ii) the NR limit of [Maxwell] ⊕ u(1) ⊕ u(1) ⊕ u(1), which is a Maxwellian generalization of the EBG.1 The use of two U (1) factors in the symmetry group was also considered in [27] in relation to AdS3/CF T2.2 ABC Z BC , and J A = 1 2 ABC J BC , and their inverse forms are J AB = ABC J C . In [32][33][34][35] the Z AB was defined as ΛZ AB where Λ is the cosmological constant. Here, instead, we would like to see the effect of including a covariant constant electromagnetic field in the three-dimensional CS gravitational system without trying to introduce a cosmological constant.In order to construct the relativistic action we will consider the most general bilinear form

We construct the first analytic examples of topologically non-trivial solutions of the (3+1)dimensional U (1) gauged Skyrme model within a finite box in (3+1)-dimensional flat space-time. There are two types of gauged solitons. The first type corresponds to gauged Skyrmions living within a finite volume. The second corresponds to gauged time-crystals (smooth solutions of the U (1) gauged Skyrme model whose periodic time-dependence is protected by a winding number). The notion of electromagnetic duality can be extended for these two types of configurations in the sense that the electric and one of the magnetic components can be interchanged. These analytic solutions show very explicitly the Callan-Witten mechanism (according to which magnetic monopoles may "swallow" part of the topological charge of the Skyrmion) since the electromagnetic field contribute directly to the conserved topological charge of the gauged Skyrmions. As it happens in superconductors, the magnetic field is suppressed in the core of the gauged Skyrmions. On the other hand, the electric field is strongly suppresed in the core of gauged time crystals.

We construct Chern-Simons gravities in (2 + 1)-dimensional space-time considering the Stringy Galilei algebra both with and without non-central extensions. In the first case, there is an invariant and non-degenerate bilinear form, however the field equations do not allow to express the spin connections in terms of the dreibeins. In the second case there is no invariant non-degenerate bilinear form. Therefore, in both cases we do not have an ordinary gravity theory. Instead, if we consider the stringy Newton-Hooke algebra with extensions as gauge group we have an invariant non-degenerate metric and from the field equations we express the spin connections in terms of the geometric fields.