Editing Covariant derivative (section)

===Remarks===
* The definition of the covariant derivative does not use the metric in space. However, for each metric there is a unique [[Torsion tensor|torsion]]-free covariant derivative called the [[Levi-Civita connection]] such that the covariant derivative of the metric is zero.
* The properties of a derivative imply that <math>\nabla_\mathbf{v} \mathbf{u}</math> depends on the values of {{mvar|u}} in a neighborhood of a point {{mvar|p}} in the same way as e.g. the derivative of a scalar function {{mvar|f}} along a curve at a given point {{mvar|p}} depends on the values of {{mvar|f}} in a neighborhood of {{mvar|p}}.
* The information in a neighborhood of a point {{mvar|p}} in the covariant derivative can be used to define [[parallel transport]] of a vector. Also the [[Curvature of Riemannian manifolds|curvature]], [[Torsion tensor|torsion]], and [[geodesic]]s may be defined only in terms of the covariant derivative or other related variation on the idea of a [[Connection (vector bundle)|linear connection]].