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## Main Question or Discussion Point

https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.damtp.cam.ac.uk/user/hsr1000/part3_gr_lectures_2017.pdf&ved=2ahUKEwi468HjtNbgAhWEeisKHRj9DNEQFjAEegQIARAB&usg=AOvVaw3UvOQyTwkcG7c7yKkYbjSp&cshid=1551081845109

Here in page 55 it is written that geodesic is a curve whose tangent vector is parallely transported along the curve.

Now if there is a curve in 2-D which is determined by λ(length along the curve) .In 2-D cartesian coordinate system the tangent vector at every λ will point along the x(unit) and y(unit) direction that means they are parallely transported along the curve that means any curve in 2-D cartesian coordinate system is a geodesic.

But the same curve in 2-D polar coordinate (r,θ) ,at each λ the tangent vector points along the r(unit) and θ(unit) which differs along the curve ,that is the same curve is not a geodesic in 2-D polar coordinate system ,is it so??

Here in page 55 it is written that geodesic is a curve whose tangent vector is parallely transported along the curve.

Now if there is a curve in 2-D which is determined by λ(length along the curve) .In 2-D cartesian coordinate system the tangent vector at every λ will point along the x(unit) and y(unit) direction that means they are parallely transported along the curve that means any curve in 2-D cartesian coordinate system is a geodesic.

But the same curve in 2-D polar coordinate (r,θ) ,at each λ the tangent vector points along the r(unit) and θ(unit) which differs along the curve ,that is the same curve is not a geodesic in 2-D polar coordinate system ,is it so??