From the definition of

covariant derivative, applied to the contravariant vector, we have

analogue to the

covariant derivative symbol [[nabla].

The theory of scale relativity aims at describing a nondifferentiable continuous manifold by the building of new tools that implement Einstein's general relativity concepts in the new context (in particular,

covariant derivative and geodesics equations).

mu]v] is the

covariant derivative, defined in analogous way as in eqs.

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].

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.

By virtue of

covariant derivative of (7) with respect to Z we get