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At any λ the tangent vector components are V

^{1}=dr(λ)/dλ along ##\hat r## and V

^{2}=dθ(λ)/dλ along ##\hat θ##.

The non-zero christoffel symbol are Γ

^{1}

_{22}and Γ

^{2}

_{12}.

From covariant derivative

(1)∇

_{1}V

^{1}=∂

_{r}V

^{1}

(2)∇

_{2}V

^{2}=∂

_{θ}V

^{2}+Γ

^{2}

_{12}V

^{1}.

(3)∇

_{2}V

^{1}=∂

_{θ}V

^{1}+Γ

^{1}

_{22}V

^{2}

(4)∇

_{1}V

^{2}=∂

_{r}V

^{2}+Γ

^{2}

_{12}V

^{2}

For this curve to be geodesic V

^{1}=1 and V

^{2}=0

And (1),(3) and (4) becomes 0 and (2)≠0.

But for a geodesic the covariant derivatives of tangent vectors are 0.

Am I missing something??

I am trying to interpret physically by taking these examples.

Thank you