Standard notation and terminology of differential geometry and general relativity are used in this article. All considerations are local so that the four-dimensional space-time M is assumed to be a smooth manifold diffeomorphic to R4. It is endowed with a metric tensor g of signature (1, 3) and a linear connection defining the covariant differentiation of tensor fields. Greek indices range from...

A reductive Lie algebra g is one that can be written C(g) ⊕ [g,g], where C(g) denotes the center of g. Equivalently, for any ideal a, there is another ideal b such that g = a⊕ b. A Cartan subalgebra of g is a subalgebra h that is maximal with respect to being abelian and having ad X being semisimple for all X ∈ h. For a reductive group, h = C(g) ⊕ h′, where h′ is a Cartan subalgebra of the semi...

A new proposal for Cartan contortion detection based on the resonant gravitational wave (GW) detector in teleparallel spacetime is considered.It is shown that Cartan contortion is proportional to the inverse of the damping time of the resonant detector.Actually the contortion acts as a damping force for the GW.A contortion of the order 10 −29 s −1 is obtained by using the data of present resona...

Abstract. The purpose of this note is to give explicit criteria to determine whether a real generalized Cartan matrix is of finite type, affine type or of hyperbolic type by considering the principal minors and the inverse of the matrix. In particular, it will be shown that a real generalized Cartan matrix is of finite type if and only if it is invertible and the inverse is a positive matrix. A...

To certain types of generic distributions (subbundles in a tangent bundle) one can associate canonical Cartan connections. Many of these constructions fall into the class of parabolic geometries. The aim of this article is to show how strong restrictions on the possibles sizes of automorphism groups of such distributions can be deduced from the existence of canonical Cartan connections. This ne...

We present an explicit numerical method to obtain the Cartan-KhanejaGlaser decomposition of a general element G ∈ SU(2 ) in terms of its ‘Cartan’ and ‘non-Cartan’ components. This effectively factors G in terms of group elements that belong in SU(2) with n < N , a procedure that an be iterated down to n = 2. We show that every step reduces to solving the zeros of a matrix polynomial, obtained b...

The interplay between topological defects, such as dislocations or disclinations, and the electronic degrees of freedom in graphene have been extensively studied. In literature, for study this kind problems, it is general used either a gauge theory curved spatial Riemannian geometry approach, where, geometric case, information about defects contained metric spin-connection. However, these can a...