Dynamics of a Homogeneous and Isotropic Space in Pure Cubic f(R) Gravity

The possible ways of dynamics of a homogeneous and isotropic space described by the Friedmann–Lemaitre–Robertson–Walker metric in the framework of cubic in the Ricci scalar f(R) gravity in the absence of matter are considered. This paper points towards an effective method for limiting the parameters of extended gravity models. A method for f(R)-gravity models, based on the metric dynamics of various model parameters in the simplest example is proposed. The influence of the parameters and initial conditions on further dynamics are discussed. The parameters can be limited by (i) slow growth of space, (ii) instability and (iii) divergence with the inflationary scenario.

Parabolic Revolution Surfaces of Finite Type in Simply Isotropic 3-spaces

In this paper, we classify parabolic revolution surfaces in the three-dimensional simply isotropic space [Formula: see text] under the condition [Formula: see text] where [Formula: see text] is the Laplace operator with respect to first and second fundamental form and [Formula: see text], [Formula: see text] are some real numbers. Also, as an application, we give some explicit examples for these surfaces.

Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces

In this work, we are interested in the differential geometry of curves in the simply isotropic and pseudo-isotropic 3-spaces, which are examples of Cayley-Klein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Euclidean and Lorentzian geometries, here we show how to build RM frames in isotropic geometries and apply them in the study of isotropic spherical curves. Indeed, through a convenient manipulation of osculating spheres described in terms of RM frames, we show that it is possible to characterize spherical curves via a linear equation involving the curvatures that dictate the RM frame motion. For the case of pseudo-isotropic space, we also discuss on the distinct choices for the absolute figure in the framework of a Cayley-Klein geometry and prove that they are all equivalent approaches through the use of Lorentz numbers (a complex-like system where the square of the imaginary unit is $+1$). Finally, we also show the possibility of obtaining an isotropic RM frame by rotation of the Frenet frame through the use of Galilean trigonometric functions and dual numbers (a complex-like system where the square of the imaginary unit vanishes).