# Why can the affinity connection be arbituarily chosen?

## Main Question or Discussion Point

In covariant derivative, we have a quantity called affinity.
The book says: When defining an affine connection, a coordinate frame must first be selected and the choice of the components of the affinity is then arbitrary within this frame. However, when these are all determined, the component of the affinity with respect to any other frame are completely fixed by the transformation law.
My question is why can the components be arbitrarily chosen?
Thank you very much!!

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Why not? An affine connection is defined to satisfy certain properties. Choosing arbitary functions as the Christoffel symbols satisfies these properties.

Why not? An affine connection is defined to satisfy certain properties. Choosing arbitary functions as the Christoffel symbols satisfies these properties.
So what is the definition of affine connection?

Am I correct to say that to define the parallel displacement such that the vector displaced does not change in magnitude and direction, we have to define F=g[ij,k] (ignore the superscript and subscript) as our affine connection?
where [ij,k]is the Christoffel symbol of the first kind and g is the metric and F is the affine connection

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