Manoff [5] applied the method of lagrangian with

covariant derivative to special type of lagrangian density depending on scalar and vector fields.

n] (n [greater than or equal to] 3) without boundary the associated scalars and the stucture tensor are such that [alpha] + b + 2c [less than or equal to] 0 and dD(X, X) [less than or equal to] 0 for all X, then a projective Killing vector field has vanishing

covariant derivative and if [alpha] + b + 2c < 0 and dD(X, X) < 0, then there does not exist any non-zero projective Killing vector field in this manifold.

The

covariant derivative of an integrable Z-connection is fully determined by its value on [M.

mu]] denotes

covariant derivative with respect to coordinates (in this case the coordinates [x.

His topics include the basics of geometry and relativity, affine connection and

covariant derivative, the geodesic equation and its applications, curvature tensor and Einstein's equation, black holes, and cosmological models and the big bang theory.

In view of the above relations, the

covariant derivative of (2.

19) Because the

covariant derivative of the metric is zero ([[nabla.

m[xi]] is the bimetric

covariant derivative with respect to [xi] and is given by [[nabla].

Let D denote the

covariant derivative with respect to G and [Mathematical Expression Omitted] the length of a tensor with respect to G, while [D.

where D is the

covariant derivative on (N, h) and [nabla] is the

covariant derivative on (M,g).

In the parentheses we recognize the prototype of the

covariant derivative.

0]) be a smooth compact d-dimensional Riemannian manifold with the Levi-Civita

covariant derivative [nabla], a Riemannian metric g, a fixed base point 0 [member of] M and a fixed orthogonal frame [u.