TORSION AND CURVATURE IN SMOOTH LOOPS

A Lie group is flat with respect to its natural left invariant affine connection, and its torsion relates to the Lie bracket via T(X, Y) = -[X, Y]. A smooth loop is, roughly, a Lie group without associativity. Its tangent algebra at the origin has, in addition to the binary bracket, a ternary operation which is a measure of deviation from associativity locally. There are several ways of giving a smooth loop affine connections. There is one for which the torsion behaves like in groups while the curvature relates to the ternary operation of the tangent algebra via R(X, Y)(Z)e = -<X(e), Y(e), Z(e)> + <Y(e), X(e), Z(e)>.