Here (the last paper in a series of eight) we end our presentation of the basics of a systematical approach to the differential geometry of smooth manifolds which uses the geometric algebras of multivector and extensors (fields) developed in previous papers. The theory of the Riemann and Ricci fields associated to a given geometric structure, i.e., a triple (M,γ, g) where M is a smooth manifold...

Considering a multi-dimensional q-oscillator invariant under the (non quantum) group U(n), we construct a q-deformed Levi-Civita epsilon tensor from the inner product states. The invariance of this q-epsilon tensor is shown to yield the quantum group SLq(n) and establishes the relationship of the U(n) invariant q-oscillator to quantum groups and quantum group related oscillators. Furthermore th...

I. Introduction-a quick historical survey of geodesic flows on negatively curved spaces. II. Preliminaries on Riemannian manifolds A. Riemannian metric and Riemannian volume element B. Levi Civita connection and covariant differentiation along curves C. Parallel translation of vectors along curves D. Curvature E. Geodesics and geodesic flow F. Riemannian exponential map and Jacobi vector fields...

In this note we briefly summarize the necessary tools from complex manifold theory in order to give an introduction to the basic results about Kähler and locally conformal Kähler manifolds. We look at many equivalent definitions of all three types of manifolds and examine in detail the parallels which arise in the theories. These parallels include the compatibility of metrics and fundamental 2-...

Radial steady-state accretion of polytropic matter is investigated under cylindrical symmetry in the Levi-Civita background metric. The model can be considered as a analog Bondi strong gravitational field. As byproduct this study, issue defining line mass density addressed and role metric free parameters discussed on example physical observables. form radial equations insensitive to structure i...

We investigate the Cartan formalism in F(R) gravity. gravity has been introduced as a theory to explain cosmologically accelerated expansions by replacing Ricci scalar R Einstein–Hilbert action with function of R. As is well-known, rewritten scalar–tensor using conformal transformation. described based on Riemann–Cartan geometry formulated vierbein-associated local Lorenz symmetry. In formalism...

Let ν be a unit normal vector field along M . Notice that ν : M −→ S satisfies that 〈ν(m),m〉 = 0. For any tangent vector v ∈ TmM , m ∈ M , the shape operator A is given by A(v) = −∇̄vν, where ∇̄ denotes the Levi Civita connection in S. For every m ∈ M , A(m) defines a linear symmetric transformation from TmM to TmM ; the eigenvalues of this transformation are known as the principal curvatures of ...

If $\psi\colon M^n\to \mathbb{R}^{n+1}$ is a smooth immersed closed hypersurface, we consider the functional $$ \mathcal{F}\_m(\psi) = \int\_M 1 + |\nabla^m \nu |^2 , d\mu, where $\nu$ local unit normal vector along $\psi$, $\nabla$ Levi-Civita connection of Riemannian manifold $(M,g)$, with $g$ pull-back metric induced by immersion and $\mu$ associated volume measure. We prove that if $m>\lflo...