Sectional curvature of connections with vectorial torsion

Riemannian manifolds of sign-defined sectional curvature have been studied by many mathematicians, due to the close relationship between curvature and the topology of Riemannian manifolds.
We investigate Riemannian manifolds whose metric connection is a connection with vectorial torsion. The Levi–Civita connection contains into this class of connections. Although the curvature tensor of these connections does not possess the symmetries of the Riemannian curvature tensor, it seems possible to determine sectional curvature. We study the question of a connectivity of the sectional curvature of connections with vectorial torsion and the sectional curvature of the Levi–Civita connection, or Riemannian curvature. We study the sign of sectional curvature of a connection with vectorial torsion. As the main test example, we consider Lie groups with a left-invariant Riemannian metric.