n] be an n-dimensional differentiable manifold endowed with a (1,1) tensor field [phi], a contravariant vector
field [xi], a covariant vector field [eta] and a Lorentzian metric g of type (0, 2) such that for each point p [member of] M, the tensor [g.
Since propagation of electromagnetic waves are always described in orthogonal frames, where there is no distinction between covariant and contravariant vector
components, the different tensor nature of vector fields ([bar.
v] is a weight 1 contravariant vector
density, its covariant divergence (with the symbol [[nabla].
Similar argument can be made for contravariant vector
mu][nu]] is symmetric in the indices [mu] and [nu], we obtain the equation of motion for the contravariant vector [v.
Such a program can be pursued by taking into account covariant and contravariant vector fields.
From the definition of covariant derivative, applied to the contravariant vector, we have
The covariant and contravariant vector fields can be also of different nature so that the above fundamental Hamiltonian invariant can be generalized as
In our picture, this means that the canonical symplectic structure is assigned in the way in which covariant and contravariant vector fields are related.
n] is a Lorentzian Para-Sasakian manifolds(briefly LP-Sasakian manifolds) if it admits a (1,1) tensor field [phi], contravariant vector
field [xi], a covariant vector field [eta], and a Lorentzian metric g, which satisfy
There the product of the balance of momentum and velocity - two contravariant vectors
o] (heavy dashed lines) compress to AB and AC (heavy solid lines), respectively, in the directions of the contravariant vectors
and the corresponding stresses are compressive.