A

tensor field A [member of] [T.sup.p.sub.q] (M) that satisfies the algebraic conditions 1), 2) of Proposition 2.1 (equivalently, satisfies (2.10)) will be called algebraically adapted to N.

An n-dimensional Lorentzian manifold M is a smooth connected paracompact Hausdorff manifold with a Lorentzian metric g, that is, M admits a smooth symmetric

tensor field 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 a non-degenerate 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.

A (2n + 1) dimensional, (n [greater than or equal to] 1) almost contact metric manifold M with almost contact metric structure ([phi], [zeta], [eta], g), where [phi] is a (1, 1)

tensor field, [zeta] is a vector field, [eta] is a 1-form and g is a compatible Riemannian metric such that

Given a pseudo-Riemannian manifold (M, g), the curvature

tensor field R of the Levi-Civita connection is called the Riemannian curvature of (M,g).

Let M be an n-dimensional almost contact metric manifold with an almost contact metric structure ([phi], [xi], [eta], g) consisting of a (1,1)

tensor field [phi], a vector field [xi] a 1-form [eta], and a Riemannian metric g on M satisfying

where I is the identity of the tangent bundle TM of M, f is a

tensor field of type (1,1), [eta] is a 1-form, [xi] is a vector field tangent to M and g is a metric

tensor field on M.

Equivalently, there is an almost contact structure ([phi], [xi], [eta]) [39] consisting of a

tensor field [phi] of type (1, 1), a vector field [xi], and a 1-form [eta] satisfying

Tensor gauge condition and

tensor field decomposition.

for all vector fields X and Y on M, where R is the Riemannian curvature of g defined by R(X,Y) = [[[nabla].sub.X], [[nabla].sub.Y]] - [[nabla].sub.[X,Y]] and [gamma](R(X,Y)) is a

tensor field of type (1,0) on T(M), which is locally expressed as [gamma](R(X, Y)) = [y.sup.s][R.sub.jks][sup.i][X.sup.j][Y.sup.k][[partial derivative].sub.[bar.i]] with respect to the induced coordinates.

If u were not null, we would have W(u, v, u, v') = 0 for any sections v, v' of [D.sup.[perpendicular to]], as one sees contracting the twice-covariant

tensor field W(* , v, * , v'), at any point x, in an orthogonal basis containing the vector [u.sub.x].

where Id is the identity

tensor field of type (1,1) on M.

Einstein's equation of motion for a photon in fields exterior to astrophysically real or imaginary spherical mass distributions who's

tensor field varies with azimuthal angle only.