Abstract
The metric connection on a space-time manifoldM defines on its tangent bundleTM a distribution of subspaces complementary to the vertical subspaces and therefore called horizontal. We give a formula for the Lie derivative with respect to the geodesic spray of the tensor field onTM which defines projection onto the vertical subspace along the horizontal subspace; and we show that this formula is a universal version of the equation, for a geodesic local vector field onM, whose trace is Raychaudhuri's equation.
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