Can I construct an affine connection on a Riemannian manifold from arbitrary Christoffel Symbols?

Question

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.

Answer 1

A connection $\nabla_X Y$ is required to be $C^\infty(M)$-linear in $X$, $\mathbb R$-linear in $Y$, and satisfy the Leibnitz rule. So if you have a coordinate system on all of $M$, or more generally, a framing $X_1, \ldots, X_n$ of the tangent bundle by some vector fields (ie linearly independent at each point of $M$), you can pick any $n^3$ functions $\Gamma^k_{ij}$ you wish, and, as you suggest, define $\nabla_{X_i}X_j=\sum_k \Gamma^k_{ij}X_k$, and extend by the above rules for arbitrary $X,Y$. If you cannot find such framing (eg for the 2 sphere), and you insist on defining a connection in this way, then you are reduced to covering $M$ by open sets on each of which you can follow yr suggestion, but then need to check that the resulting connection is consistently defined on the intersections of the open sets of your cover. So you need formulas that tells you how the $\Gamma$'s transform when the framing is changed. These you can derive easily from the above axioms for a connection.