Editing Covariant derivative (section)

==Derivative along a curve==
Since the covariant derivative <math>\nabla_X T</math> of a tensor field {{mvar|T}} at a point {{mvar|p}} depends only on the value of the vector field {{mvar|X}} at {{mvar|p}} one can define the covariant derivative along a smooth curve <math>\gamma(t)</math> in a manifold:
<math display="block">D_tT=\nabla_{\dot\gamma(t)}T.</math>
Note that the tensor field {{mvar|T}} only needs to be defined on the curve <math>\gamma(t)</math> for this definition to make sense.

In particular, <math>\dot{\gamma}(t)</math> is a vector field along the curve <math>\gamma</math> itself. If <math>\nabla_{\dot\gamma(t)}\dot\gamma(t)</math> vanishes then the curve is called a geodesic of the covariant derivative. If the covariant derivative is the [[Levi-Civita connection]] of a [[Metric tensor|positive-definite metric]] then the geodesics for the connection are precisely the [[geodesics]] of the metric that are parametrized by [[Arc length#Generalization to (pseudo-)Riemannian manifolds|arc length]].

The derivative along a curve is also used to define the [[parallel transport]] along the curve.

Sometimes the covariant derivative along a curve is called '''absolute''' or '''intrinsic derivative'''.