Sectional Curvature of Connections with Vectorial Torsion

Riemannian manifolds of sign-definite sectional curvature have been studied by many mathematicians due to the close relationship between the curvature and the topology of a Riemannian manifold. We study Riemannian manifolds whose metric connection is a connection with vectorial torsion. The class of such connections contains the Levi-Civita connection. Although the curvature tensor of such a connection does not possess symmetries of the curvature tensor of the Levi-Civita connection, it is possible to define the sectional curvature. We investigate the question on relations between the sectional curvature of a connection with vectorial torsion and the sectional curvature of the Levi-Civita connection (Riemannian curvature). We also study the sign of sectional curvatures of connections with vectorial torsion. As an example, we consider Lie groups with left-invariant Riemannian metrics.