We discuss some aspects of the differential geometry of curves in Minkowski space. We establish the Serret-Frenet equations in Minkowski space and use them to give a very simple proof of the fundamental theorem of curves in Minkowski space. We also state and prove two other theorems which represent Minkowskian versions of a very known theorem of the differential geometry of curves in tridimensi...

A nonassociative algebra endowed with a Lie bracket, called a torsion algebra, is viewed as an algebraic analog of a manifold with an affine connection. Its elements are interpreted as vector fields and its multiplication is interpreted as a connection. This provides a framework for differential geometry on a formal manifold with a formal connection. A torsion algebra is a natural generalizatio...

Let G be a complex simply connected semisimple Lie group and let Γ torsionless uniform irreducible lattice in G. Then Γ﹨G is compact non-Kähler manifold whose tangent bundle holomorphically trivial. In this note we compute the Picard of when rank(G)≥3. When rank(G)<3, determine Pic0(Γ﹨G)⊂Pic(Γ﹨G) topologically trivial holomorphic line bundles. rank(G)≥2, also show that Pic0(PΓ) isomorphic to Pi...

We consider the sum of Einstein-Hilbert action and a Pontryagin density (PD) in arbitrary even dimension $D$. All curvatures are functions independent affine (torsionless) connections only. In dimension, not only $D=4n$, these first order PD terms shown to be covariant divergences "Chern-Simons" currents. The field equation for connection leads it being Levi-Civita, metric equations equivalent ...

It has been recently shown that, in the first order (Palatini) formalism, there is universality of Einstein equations and Komar energy– momentum complex, in the sense that for a generic nonlinear Lagrangian depending only on the scalar curvature of a metric and a torsionless connection one always gets Einstein equations and Komar’s expression for the energy–momentum complex. In this paper a sim...

The classical dynamics of N spinning point sources in 2+1 Einstein-Cartan gravity is considered. It corresponds to the ISO(2, 1) Chern-Simons theory, in which the torsion source is restricted to its intrinsic spin part. A class of explicit solutions is found for the dreibein and the spin connection, which are torsionless in the spinless limit. By using the residual local Poincaré invariance of ...

Abstract Let ( M , ∇, 〈 〉 ) be a manifold endowed with flat torsionless connection r and Riemannian metric T k ≥1 the sequence of tangent bundles given by = −1 1 TM . We show that, for any ≥ 1, carries Hermitian structure J g ∇ when is Lie group (∇, are left invariant there on each such that invariant. It well-known Kähler if only Hessian, i.e, in system affine coordinates x ., n ), <m:math xml...

In Poincaré gauge theory of gravity and in Poincaré gauge theory of gravity, we give the necessary and sufficient condition in order that the Schwarzschild spacetime expressed in terms of the Schwarzschild coordinates is obtainable as a torsionless exact solution of gravitational field equations with a spinless point-like source having the energy-momentum density T̃ ν μ (x) = −Mc2δ 0 μ δ ν 0 δ(3...