Connections with parallel skew-symmetric torsion on sub-Riemannian manifolds

Abstract

On a sub-Riemannian manifold M of contact type, is considered an N-connection defined by the pair (, N), where is an interior metric connection, is an endomorphism of the distribution D. It is proved that there exists a unique N-connection such that its torsion is skew-symmetric as a contravariant tensor field. A construction of the endomorphism corresponding to such connection is found. The sufficient conditions for the obtained connection to be a metric connec­tion with parallel torsion are given.

Lorentzian connections with parallel twistor-free torsion

“…Thus we may assume that the geometry is indecomposable. In Section 3 from [12] we proved that if ∇ is a metric connection on a Lorentzian manifold (N, g) with parallel skew-symmetric torsion τ and weakly irreducible holonomy algebra, then τ automatically satisfies the condition σ τ (X) = 0, moreover, (N, g) is as in the case 1 or 2 from the statement of the theorem. Now we assume that the geometry (N, g, ∇) is reducible and indecomposable.…”

“…From Lemma 1 it follows that τ is ∇-parallel. In [12] we described holonomy, curvature and torsion of Lorentzian connections with parallel skew-symmetric torsion.…”

“…The geometry (N, g, ∇) is called decomposable if the holonomy algebra g ⊂ so(1, n + 1) preserves an orthogonal decomposition (12) such that it holds…”

“…Now we assume that the geometry (N, g, ∇) is reducible and indecomposable. Then the holonomy algebra g preserves a decomposition (12). We may assume that the induced representation of g in L is weakly irreducible.…”

“…Connections with skew-symmetric torsion appear in certain supergravity theories, see, e.g., [10,11,15,19]. We studied Lorentzian metric connections with parallel skew-symmetric torsion in [12].…”