Someone correctly computes the Christoffel symbols of the Euclidean plane in plane…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch The metric, the connection and geodesics

Problem

Someone correctly computes the Christoffel symbols of the Euclidean plane in plane polar coordinates,

\Gamma^r{}_{\theta\theta} = -r

and

\Gamma^\theta{}_{r\theta} = \Gamma^\theta{}_{\theta r} = 1/r

, all others zero, then argues: "These are not all zero, so the plane must be curved." ((a)) Find the error in this reasoning. ((b)) Name a coordinate system on the very same plane in which the symbols do all vanish, and explain why that does not make the plane curved in polars and flat in that system.

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