Let [M.sup.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.sub.p]: [T.sub.p]M x [T.sub.p]M [right arrow] R is an inner product of signature (-, +, +, ..., +), where [T.sub.p]M denotes the tangent vector space of M at p and R is the real number space which satisfies
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.E] and [bar.D]) remains masked (the same for [bar.B] and [bar.H]).
A vector (called a contravariant vector
) represents a point in the Euclidean space, or primary space, from the origin to the point; A vector in a 3-space is represented by:
Precisely because [[summation].sub.v] [T.sup.[mu]v] [square root of -g] [[xi].sub.v] is a weight 1 contravariant vector
density, its covariant divergence (with the symbol [[nabla].sub.[mu]]) equals its coordinate divergence (with the symbol [partial derivative]/[partial derivative][x.sup.[mu]]) and is a scalar density of weight 1.
Similar argument can be made for contravariant vector
Furthermore, the contravariant vector [u.sup.[mu]] with components u = ([u.sup.0], [u.sup.1], [u.sup.2], [u.sup.3]) is the four-velocity
(28) and (29) and the fact that the affine connection [[GAMMA].sup.[lambda].sub.[mu][nu]] is symmetric in the indices [mu] and [nu], we obtain the equation of motion for the contravariant vector [v.sup.[lambda]]
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
An n-dimensional differentiable manifolds [M.sup.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
Figure 6 shows that the original lines [AB.sub.o], and [AC.sub.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.