properties of regular partial derivatives also apply to covariant derivatives: In general relativity, one usually refers to "the" covariant derivative, which is the one associated with Levi-Civita affine connection. By definition, Levi-Civita connection preserves the metric under parallel transport, therefore, the covariant derivative gives zero when acting on a metric tensor (as well as its inverse). It means that we can take the (inverse) metric tensor in and out of the derivative and use it to raise and lower indices: The Lie derivative Another important tensorial derivative is the Lie derivative. Unlike the covariant derivative, the Lie derivative is independent of the metric, although in general relativity one usually uses an expression that seemingly depends on the metric through the affine connection. Whereas the covariant derivative required an affine connection to allow comparison between vectors at different points, the Lie derivative uses a congruence from a vector field to achieve the same purpose. The idea of Lie dragging a function along a congruence leads to a definition of the Lie derivative, where the dragged function is compared with the value of the original function at a given point. The Lie derivative can be defined for type tensor fields and in this respect can be viewed as a map that sends a type to a type tensor. The Lie derivative is usually denoted by , where is the vector field along whose congruence the Lie derivative is taken. The Lie derivative of any tensor along a vector field can be expressed through the covariant derivatives of that tensor and vector field. The Lie derivative of a scalar is just the directional derivative: Higher rank objects pick up additional terms when the Lie derivative is taken. For example, the Lie derivative of a type tensor is More generally, In fact in the above expression, one can replace the covariant derivative with any torsion free connection or locally, with the coordinate dependent derivative , showing that the Lie derivative is independent of the metric. The covariant derivative is convenient however because it commutes with raising and lowering indices. One of the main uses of the Lie derivative in general relativity is in the study of spacetime symmetries where tensors or other geometrical objects are preserved. In particular, Killing symmetry (symmetry of the metric tensor under Lie dragging) occurs very often in the study of spacetimes. Using the formula above, we can write down