Why Curvature Tensor in Bianchi identity is not all zero

riemannian-geometrytensors

I am reading Tensor Calculus and Differential Geometry. In chapter4 [ Riemannian Geometry ] under differentail property of Covariant curvature tensor in property 4.1.7.

Where Riemann Christoffel Tensor is given as
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Now, It is mentioned that When any Pole P is choosen then Christoffel Symbols vanish in that pole [ from property of Christoffel Symbols ] as below

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which is true but then it is

and Result will be
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Now I was wondering why the first and second term also not zero as these also have Christoffel Symbols term
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Then second derivatives multiplied by this term should result zero and total sum must be zero.why is not like that.

Thanks in advance.

Best Answer

Your question is why a choice that ensures the Christoffel symbols vanish locally doesn't have the same effect on their second derivatives. It comes down to behaviour in a neighbourhood. To take a much simpler example, $x^3-x^2$ vanishes at $x=0$, as does its first derivative $3x^2-2x$, but the second derivative, $6x-2$, doesn't.

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