The question is rather simple. All my definitions are as in Do Carmo's "Riemannian Geometry".
If $M$ is a Riemannian Manifold, can I construct an affine connection $\nabla$ on it by setting, for all $i,j=1,...,n$ $$ \nabla _{\partial x_i} \partial x_j=\sum \Gamma^k_{ij}\partial x_k $$ where $\Gamma^k_{ij}$, $i,j,k=1,...,n=dim(M)$ are arbitrary $C^\infty$ functions over $M$? If not, is there any way I can stablish conditions for the symbols for this construction to work?
I see no problem in doing so. The Christoffel symbols does not seem to depend on the metric, neither am I requiring the connection to be symmetric or compatible with the metric.