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
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
which is true but then it is
Now I was wondering why the first and second term also not zero as these also have Christoffel Symbols term
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.