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- Covariant derivative of the Riemann tensor evaluated in Riemann normal coordinates

Hello everyone,

in equation 3.86 of this online version of Carroll´s lecture notes on general relativity (https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html) the covariant derviative of the Riemann tensor is simply given by the partial derivative, the terms carrying the Christoffel symbols seem to vanish. I assume this results from the note that it is evaluated in Riemann normal coordinates. I know one can choose the coordinate system so that a given Christoffel-symbol vanishes but in this case there are so many to handle so that I am not convinced this is working. Can someone please give me an input on how to make myself clear that it works? Thanks in advance!

in equation 3.86 of this online version of Carroll´s lecture notes on general relativity (https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html) the covariant derviative of the Riemann tensor is simply given by the partial derivative, the terms carrying the Christoffel symbols seem to vanish. I assume this results from the note that it is evaluated in Riemann normal coordinates. I know one can choose the coordinate system so that a given Christoffel-symbol vanishes but in this case there are so many to handle so that I am not convinced this is working. Can someone please give me an input on how to make myself clear that it works? Thanks in advance!