Every possible choice of Christoffel symbols generate a valid connection

Question

Does every possible choice of Christoffel symbols generate a valid connection? Or is there some restriction on them?

Answer 1

The space of affine connections is an affine space, not a vector space. The difference of two affine connections is a tensor $A_{ij}\,^{k}$ of type $\tbinom{1}{2}$; so the space of connections is an affine space modeled on the vector space of such tensors. Another way to say this is that Christoffel symbols do not transform tensorially but the difference of two connections does. As a consequence, one has to be careful with what one means by "every possible choice of Christoffel symbols". If one fixes a particular connection $\nabla$ as a reference, then the affine space of connections becomes a vector space with the reference connection as the origin (for example, one can fix coordinates $x^{1}, \dots, x^{n}$, and choose as the reference connection the connection such that the coordinate coframe $dx^{1}, \dots, dx^{n}$ is parallel; this connection is what is usually written as the partial derivative operator). For any tensor $A_{ij}\,^{k}$, $\nabla + A$ is another connection. If $\Gamma_{ij}\,^{k}$ are the Christoffel symbols of $\nabla$ with respect to some frame, then $\Gamma_{ij}\,^{k} + A_{ij}\,^{k}$ are the Christoffel symbols of $\nabla + A$ with respect to this same frame. In this sense, every possible choice of Christoffel symbols generates a valid connection. (If one wishes to speak only of torsion-free connections, then one needs to restrict attention to the vector space of tensors $A_{ij}\,^{k}$ symmetric in the lower indices.)