Manoff  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
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
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.