Why can the affinity connection be arbituarily chosen?

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The discussion centers on the concept of affine connections in the context of covariant derivatives. It establishes that while the components of the affinity can be arbitrarily chosen within a selected coordinate frame, they become fixed when transformed to another frame according to the transformation law. The arbitrary selection of Christoffel symbols is permissible as long as the defined affine connection satisfies specific properties. Furthermore, the relationship between the affine connection, the Christoffel symbols, and the metric tensor is clarified, emphasizing the necessity of defining the connection to maintain vector magnitude and direction during parallel displacement.

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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.
 
zhentil said:
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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