In this short note we study flat metric connections with antisymmetric torsion T 6= 0. The result has been originally discovered by Cartan/Schouten in 1926 and we provide a new proof not depending on the classification of symmetric spaces. Any space of that type splits and the irreducible factors are compact simple Lie group or a special connection on S. The latter case is interesting from the ...

We consider spacetime to be a connected real 4-manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is quadratic in curvature and study the resulting system of Euler–Lagrange equations. In the first part of the paper we lo...

Theorem 1. 1) Under local changes of the fields u = u(w) the coefficient g(u) in the bracket (2) transforms like a bilinear form (a tensor with upper indices); if det g 6= 0, then the expression b k (u) = gΓ j sk transforms in such a way that the Γjsk are the Christoffel tymbols of a differential-geometric connection. 2) In order that the bracket (2) be skew-symmetric it is necessary and suffic...

The torsion of every metric connection on a Riemannian manifold has three components: one totally skew-symmetric, vectorial type and twistorial type, which is also called the traceless cyclic component. In this paper we classify complete simply connected manifolds carrying whose parallel, nonzero component vanishing

In this work, the cases of non-integrable distributions in a Riemannian manifold with first generalized semi-symmetric non-metric connection and second are discussed. We obtain Gauss, Codazzi, Ricci equations both cases. Moreover, Chen’s inequalities also obtained Some new examples based on connections proposed.