Modeling Thermodynamic Distance, Curvature and Fluctuations by Viorel Badescu

Author:Viorel Badescu
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

2. ;

3. , where is the ordinary directional derivative in the direction of A. This last property is called Leibnitz rule .

One defines the parallel transport of a vector V along a curve , whose tangent vector field is denoted , as the (unique) vector field along having the following properties1. ;

2. along .

The notion of covariant derivative immediately follows: the covariant derivative of V along is given by the vector field

(4.145)

Based on the Eq. (4.145) (and forcing the language) it is often said about that it is the covariant derivative of Y along X, where X and Y are two arbitrary vector fields. For a given metric g, among all possible linear connections, there is one and only one, with the following properties:(i)it is symmetric, i.e.

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