Derivative Along Curve
Since the covariant derivative of a tensor field at a point depends only on value of the vector field at one can define the covariant derivative along a smooth curve in a manifold:
Note that the tensor field only needs to be defined on the curve for this definition to make sense.
In particular, is a vector field along the curve itself. If vanishes then the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a certain metric then the geodesics for the connection are precisely the geodesics of the metric that are parametrised by 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.
Read more about this topic: Covariant Derivative
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