Kinematics of vector fields
  Different (not only by sign) affine connections are introduced for contravariant and covariant tensor fields over a differentiable manifold by means of a non-canonical contraction operator, defining the notion dual bases and commuting with the covariant and with the Lie-differential operator. Classification of the linear transports on the basis of the connections between the connections is given. Notion of relative velocity and relative acceleration for vector fields are determined. By means of these kinematic characteristics several other types of notions as shear velocity, shear acceleration, rotation velocity, rotation acceleration, expansion velocity and expansion acceleration are introduced and on their basis auto-parallel and non-isotropic (non-null) vector fields are classified. 
S. Manoff