If any set of points is displaced by where all distance relationships are unchanged (i.e., there is an isometry), then the vector field is called a Killing vector.
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so let
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where is the Lie derivative.
An ordinary derivative can be replaced with a covariant derivative in a Lie derivative, so we can take as the definition
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which gives Killing's equation
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where denotes the symmetric tensor part and is a covariant derivative.
A Killing vector satisfies
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where is the Ricci curvature tensor and is the Riemann tensor.
In Minkowski space, there are 10 Killing vectors
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The first group is translation, the second rotation, and the final corresponds to a "boost."