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Suppose that $V$ is a vector field tangent to a family of geodesic curves. Furthermore suppose that there exists another vector field $T$ , where T is the tangent vector field that determines how far the geodesics are from each other. I know how to compute an expression such as $\nabla_VV$ however I do not know how to compute an expression such as $\nabla_T\nabla_VV$. How would I compute this second covariant derivative?

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  • $\begingroup$ You have $\nabla_V V = \mu V$ for some $\mathbb C$-valued function $\mu$ (which is zero if you take affine parametrization). So $\nabla_T \nabla_V V = \nabla_T (\mu \nabla V) $ ($=0$ with affine parametrization) $\endgroup$ Oct 19 '21 at 7:58
  • $\begingroup$ I know that the expression equals 0 but how can I show that it equals 0? $\endgroup$
    – aygx
    Oct 19 '21 at 12:19
  • $\begingroup$ By definition, if the geodesic have an affine parameter (such as proper time in GR), $\nabla_V V = 0$. So its derivative (covariant or not) must also be zero. $\endgroup$ Oct 19 '21 at 21:36

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